The visitors were far from their best in a scrappy game and missed a late penalty when Cerny saved from Marc-Antoine Fortune.
But the most important thing for Tony Mowbray and the nervy Celtic support was that three points were claimed.
Once again the Bhoys boss had tinkered with his line-up and once again, they were far from convincing.
Jos Hooiveld made his debut, the former AIK Solna player teaming up with 18-year-old Josh Thompson at the heart of the Hoops defence as Landry NGuemo, back from international duty with Cameroon at the African Nations Cup, returned to the midfield.
Ki Sung-yueng came in for Marc Crosas, who dropped out of the starting XI following the 2-1 midweek loss to Hibernian and was named on the bench alongside Niall McGinn and Paul Caddis.
Brian Easton made a return to a re-vamped Hamilton side after agreeing a loan deal from Burnley, the team he joined from Accies just six months ago.
Joel Thomas, who rejoined Billy Reid's side on loan from Colchester on Friday, was also handed a start.
Both sides appeared disjointed at times throughout a scrappy first 45 minutes which began with a couple of early scares for Celtic.
In the fourth minute Hoops defender Andreas Hinkel had to head an Easton corner off the line before the visitors smuggled the ball to safety.
Moments later, Hamilton midfielder Jordan Kirkpatrick almost met a looping Trent McClenahan cross from the right.
Celtic slowly worked their way into the game and Accies defender Mark McLaughlin nearly turned the ball into his own net before Aiden McGeady went close with a drive from the edge of the box.
Thomas was working hard against Hooiveld and Thompson but was finding little support from his team-mates.
Hinkel threw a couple of deep crosses into the Accies box from the right but to no avail and then Fortune burrowed his way into the penalty area but failed to test Cerny with his shot.
Nine minutes from the break McLaughlin intercepted a pass to Georgios Samaras then strode purposefully towards goal only to see his 30-yard drive speed wide.
Then, in a Celtic break, Cerny blocked Fortune's shot before the striker ballooned the rebound over the bar.
Hoops captain Darren O'Dea headed a Ki corner over the bar before Hooiveld hit the side netting following another corner from the Korean seconds later.
Celtic started the second half in more determined mood.
Within minutes of the restart Ki sent O'Dea racing down the left but when the Irishman's cross came in to the box, Accies defender David Elebert cleared the danger.
Seconds later a deflected NGuemo drive from the edge of the box flew in to the arms of Cerny.
In the 51st minute, McClenahan was replaced by former Celtic and St Mirren defender David van Zanten but the stalemate looked set to continue.
On the hour mark, after Samaras set him up eight yards out, Ki sent a weak header limping past the far post.
Five minutes later Zheng Zhi and Samaras were replaced by McGinn and Rasmussen, seconds before Ki drew a fine save from Cerny with a powerful drive from just inside the box.
But in the 67th minute it was Rasmussen who broke the deadlock when he turned and drove in from close range, after the ball had fallen kindly to him when Ki had been blocked.
The former Brondby striker was then booked by referee Calum Murray for over-celebrating with the travelling fans.
Ali Crawford came on for Easton with 10 minutes remaining as Accies tried to get back into the game which was stretching by the minute.
Cerny had to make a near-post save from Fortune after the former Nancy striker was set up by McGinn.
Hamilton threw everything forward in the final minutes but never looked like levelling.
In the first minute of stoppage time McLaughlin was adjudged to have fouled Fortune in the box to concede a penalty but Cerny made a great save from the forward.