Celtic moved three points clear at the top of the Scottish Premiership, helped by two goalkeeping howlers from Hamilton Academical’s Ryan Fulton.
The former Liverpool youth had denied the reigning champions with a series of first-half saves.
But Fulton allowed a Callum McGregor drive to slip through his legs.
Then, after the break, a McGregor cross bounced off his chest and Ryan Christie was on hand to tap home before Scott Sinclair headed a late third.
Hamilton had lost their last eight meetings with Celtic – and their previous six visits to the Scottish champions’ home – and, despite some stubborn resistance, they at no point looked like ending that trend.
As Accies suffered their fifth consecutive away defeat, Brendan Rodgers’ side notched up the same number of home wins since losing to RB Salzburg – their only Celtic Park reverse this season.
Mixed bag for goalkeeper Fulton
If third-bottom Hamilton were to have avoided an expected defeat, they would have needed a hero. It looked like it may have been Fulton.
He made half a dozen or so decent saves in the first half as Celtic peppered their opponents’ goal with shot after shot.
The standout save came from Sinclair. The winger drove into the box and fired off a shot that Fulton brilliantly touched on to the crossbar.
But football can be a cruel game and, after all his good work, the keeper would be undone by his own bad mistake.
McGregor fired in a shot from long range that looked like it would be comfortably taken, but this time the man in the Accies goal fumbled and the ball trundled between his legs and into the net.
On the balance of chances, the goal was deserved for Celtic, albeit they never really hit top gear.
There was more agony for Hamilton as Tony Andreu poked his shot wide when handed a chance to equalise on the stroke of half-time.
Celtic’s dominance would continue in the second half, but they were wasteful and they almost paid the price when Steven Boyd drove through the middle and his shot was diverted just wide by goalkeeper Scott Bain.
Another moment to forget for Fulton would decide the outcome when he spilled McGregor’s cross and it fell kindly for Christie.
Sinclair put the icing on Rodgers’ birthday cake – the manager today turned 46 – with a clinical header.
As you were at top and bottom – analysis
A group of Celtic fans unfurled a banner calling on the club board to back Rodgers or risk losing him.
But the manager – before the match – was keen to stress there was nothing in any speculation of a fallout between himself and the club hierarchy.
On Friday, he had also said he would be happy to go with the players he has got.
Nevertheless, as January draws to a close, attention will focus on what business – if any – Celtic do in the transfer market.
On the field, a double-header against St Johnstone is next on the agenda – at home on Wednesday then away on Sunday.
Next up for Hamilton is a cut-throat relegation battle next weekend at home to Dundee, who remain a point behind after their 1-0 defeat at home to Motherwell.
‘It was important to win well’
Celtic manager Brendan Rodgers said: “I am very pleased [with the performance]. It was important for us to win, and to win well and again I thought it was another good game for us.
“The pitch is obviously a bit slower with the issues it has so that slows our game a fraction, but I thought a lot of our combination play and the structure of our game against this kind of team, was very pleasing.”
Hamilton boss Martin Canning said of Ryan Fulton: “That’s the life of a young goalkeeper. For a large part of that game he was brilliant, he made so many good saves. But unfortunately for him when you make mistakes as a goalkeeper it will usually cost a goal.
“He’s got to make sure his mentality is right and he gets on with it because he showed for a large part of that game how good a goalkeeper he is.”