Prior to Rasmussen's strike, Celtic had toiled to break down stubborn, if unadventurous, opponents, with little goalmouth action to excite the fans.
Celtic were awarded a late penalty but Tomas Cerny made a good save to deny Marc-Antoine Fortune.
It was a hard slog for the visitors, who rarely found any passing fluency on a badly rutted pitch.
Visiting manager Tony Mowbray will be pleased that Rasmussen opened his account but the lack of cutting edge and midfield creativity must be a concern for the team chasing league leaders Rangers.
The home side made a lively start, with Celtic scrambling to clear a dangerous fourth-minute corner from Brian Easton, while Jordan Kirkpatrick was inches away from connecting with a Trent McClenahan cross.
However, the Accies completed the 90 minutes without a serious effort on target.
Celtic slowly worked their way into the game and Hamilton defender Mark McLaughlin awkwardly turned the ball over his own crossbar before Aiden McGeady went close with a drive from 18 yards.
Celtic full-back Andreas Hinkel was getting forward frequently but his team-mates could not capitalise on the German's good deliveries into the penalty area.
Nine minutes from the break, McLaughlin stepped in to intercept a slack pass to Georgios Samaras then strode forward purposefully only to see his 30-yard drive fly wide.
Fortune broke free of the home defence but his shot was well blocked by Cerny before the striker ballooned the rebound over the bar.
Celtic skipper Darren O'Dea headed over from a corner before Jos Hooiveld hit the side-netting following another set-piece from Ki Sung-Yueng.
Early in the second half Landry N'Guemo made a powerful run from the heart of midfield but his shot took a deflection and looped into the arms of Cerny.
On 65 minutes, South Korean midfielder Ki did well to manufacture room for shot on the edge of the penalty area but his powerful effort was beaten away by Cerny.
The breakthrough came two minutes later when Rasmussen, on as a replacement for the ineffectual Samaras, managed to turn sharply and shoot low into the net after a strong run from Ki.
The visitors were knocking the ball around with more purpose after the goal but still found it difficult to prise open the Accies defence.
A fierce angled drive from Fortune was blocked at his near post by Cerny and the keeper again denied the Celtic striker from a penalty in time added on.
Fortune was bundled over by McLaughlin but his spot-kick lacked accuracy and Cerny was able to parry the ball to safety.
Live text and stats
Saturday, 30 January 2010
Scottish Premier
Home Team
Score
Away Team
Time
Hamilton
0-1
Celtic
R
(HT 0-0)
Rasmussen 67
Hamilton
01 Cerny
02 McClenahan (van Zanten 51)
04 Canning
06 McLaughlin
15 Easton (Crawford 81)
25 Elebert
08 McArthur
27 Mensing
33 Wilkie
26 Thomas
28 Kirkpatrick (Marco Paixao 70)
Substitutes
19 Murdoch,
03 Hastings,
05 van Zanten,
36 Crawford,
14 Elliott,
18 Flavio Paixao,
21 Marco Paixao
Celtic
01 Boruc
02 Hinkel
16 Hooiveld
38 Thompson
48 O'Dea
06 N'Guemo
18 Ki
27 Zhi (McGinn 64)
46 McGeady (Caddis 85)
09 Samaras (Rasmussen 64 yellow card)
10 Fortune
Substitutes
24 Zaluska,
52 Caddis,
14 McGinn,
17 Crosas,
20 McCourt,
19 Rasmussen,
55 McGowan
Ref: MurrayAtt: 4,922
HAMILTON CELTIC
Possession
Hamilton 47%
Celtic 53%
Attempts on target
Hamilton 1
Celtic 6
Attempts off target
Hamilton 3
Celtic 7
Corners
Hamilton 5
Celtic 6
Fouls
Hamilton 7
Celtic 9
Final Result
Full Time
90:00+3:48 The match has reached full-time.
90:00+0:31 Penalty taken by Marc-Antoine Fortune is saved by Tomas Cerny. Hamilton 0-1 Celtic.
90:00+0:31 Penalty awarded for a foul by Mark McLaughlin on Marc-Antoine Fortune.
87:24 Kyle Wilkie takes the outswinging corner, clearance by Landry N'Guemo.
86:22 Ali Crawford fouled by Landry N'Guemo, the ref awards a free kick. Free kick taken by Kyle Wilkie.
84:46 Free kick taken by Artur Boruc.
84:46 Substitution Paul Caddis is brought on as a substitute for Aiden McGeady.
84:46 Handball by Joel Thomas.
82:52 Marc-Antoine Fortune takes a shot. Save by Tomas Cerny.
80:26 Substitution Ali Crawford is brought on as a substitute for Brian Easton.
79:22 Marc-Antoine Fortune is penalised for handball and concedes a free kick. Brian Easton takes the free kick.
76:05 Foul by Joel Thomas on Sung-Yong Ki, free kick awarded. Andreas Hinkel takes the free kick.
73:57 Free kick awarded for a foul by Morten Duncan Rasmussen on Kyle Wilkie. David Elebert takes the direct free kick.
73:15 The official flags Joel Thomas offside. Artur Boruc takes the free kick.
72:21 Free kick awarded for a foul by Morten Duncan Rasmussen on Brian Easton. Direct free kick taken by David Elebert.
71:26 Morten Duncan Rasmussen is flagged offside by the assistant referee. Indirect free kick taken by Tomas Cerny.
70:46 Niall McGinn takes a shot. Save by Tomas Cerny.
69:53 Mark McLaughlin gives away a free kick for an unfair challenge on Marc-Antoine Fortune. The free kick is delivered right-footed
by Sung-Yong Ki from right wing, clearance by David Elebert.
69:13 Substitution Marco Paixao joins the action as a substitute, replacing Jordan Kirkpatrick.
67:45 Booking The referee shows Morten Duncan Rasmussen a yellow card.
66:44 GOAL - Morten Duncan Rasmussen:Hamilton 0 - 1 Celtic Goal scored by Morten Duncan Rasmussen from close in to the bottom left corner of the goal. Hamilton 0-1 Celtic.
64:47 Drilled left-footed shot by Sung-Yong Ki. Tomas Cerny makes a save.
63:05 Substitution Morten Duncan Rasmussen on for Georgios Samaras.
63:05 Substitution Niall McGinn joins the action as a substitute, replacing Zheng Zhi.
60:28 Free kick awarded for a foul by Landry N'Guemo on Joel Thomas. Direct free kick taken by Brian Easton.
59:06 Sung-Yong Ki produces a headed effort from deep inside the six-yard box which goes wide of the right-hand upright.
58:00 Corner taken left-footed by Brian Easton, Josh Thompson makes a clearance.
55:55 Corner taken by Sung-Yong Ki, save by Tomas Cerny.
55:23 Corner taken right-footed by Sung-Yong Ki.
50:50 Substitution David van Zanten on for Trent McClenahan.
50:11 Landry N'Guemo takes a shot. Save made by Tomas Cerny.
48:02 James McArthur fouled by Landry N'Guemo, the ref awards a free kick. David Elebert takes the direct free kick.
46:49 Joel Thomas has an effort at goal from outside the penalty box missing to the wide left of the goal.
45:01 The referee starts the second half.
Half Time
45:00+2:19 The whistle is blown to end the first half.
44:24 Jordan Kirkpatrick takes a shot from just outside the penalty area which clears the crossbar.
42:47 Corner taken by Sung-Yong Ki from the left by-line to the near post, Jos Hooiveld has an effort at goal from deep inside the
area missing to the left of the goal.
41:27 Sung-Yong Ki takes the inswinging corner, Headed effort from deep inside the penalty area by Darren O'Dea goes harmlessly
over the crossbar.
40:27 Unfair challenge on Kyle Wilkie by Zheng Zhi results in a free kick. Jordan Kirkpatrick produces a strike on goal direct from
the free kick.
39:50 Inswinging corner taken left-footed by Brian Easton from the right by-line, save by Artur Boruc.
38:51 Marc-Antoine Fortune takes a shot. Save by Tomas Cerny. Marc-Antoine Fortune takes a shot from just inside the penalty box
which clears the bar.
37:40 Corner taken right-footed by Sung-Yong Ki, clearance made by Simon Mensing.
36:58 Mark McLaughlin challenges Aiden McGeady unfairly and gives away a free kick. Andreas Hinkel takes the direct free kick.
35:58 Mark McLaughlin has an effort at goal from outside the box which goes wide left of the target.
32:56 Aiden McGeady produces a left-footed shot from deep inside the area which goes wide of the left-hand post.
29:04 Brian Easton fouled by Aiden McGeady, the ref awards a free kick. Free kick taken by Brian Easton.
21:34 The referee blows for offside. Tomas Cerny takes the direct free kick.
18:42 Unfair challenge on Mark McLaughlin by Georgios Samaras results in a free kick. David Elebert takes the direct free kick.
18:08 Landry N'Guemo produces a left-footed shot from just outside the area that goes harmlessly over the target.
14:41 Jordan Kirkpatrick gives away a free kick for an unfair challenge on Andreas Hinkel. Landry N'Guemo takes the free kick.
14:05 Aiden McGeady sends in a cross, save by Tomas Cerny.
11:37 Aiden McGeady produces a left-footed shot from just outside the box that goes wide right of the target.
11:13 Sung-Yong Ki decides to take a short corner.
7:43 The offside flag is raised against Marc-Antoine Fortune. Indirect free kick taken by Tomas Cerny.
6:06 Simon Mensing concedes a free kick for a foul on Marc-Antoine Fortune. Free kick taken by Sung-Yong Ki.
5:23 Foul by Joel Thomas on Josh Thompson, free kick awarded. Direct free kick taken by Artur Boruc.
4:58 Brian Easton fouled by Sung-Yong Ki, the ref awards a free kick. Direct free kick taken by James McArthur.
3:47 Corner taken left-footed by Brian Easton.
2:43 Inswinging corner taken from the right by-line by Brian Easton, clearance by Andreas Hinkel.
0:43 Direct free kick taken by Tomas Cerny.
0:00 The referee gets the match started.
Live text and data provided by The Press Association.
Source: BBC Sport
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