Attacker Ryan Christie pounced to score from close range in the 13th minute – his fourth in three games – but the home side were far from their best.
A dogged Accies side continued to make life uncomfortable for Celtic and it looked like they would snatch a point when substitute Marios Ogkmpoe bundled in an equaliser in the 89th minute.
However, in the second of three added minutes, skipper Brown fired in a drive for a crucial victory.
With Rangers letting a two-goal lead slip to draw 2-2 with Aberdeen at Pittodrie, Celtic are now two points clear of the Light Blues ahead of their Betfred Cup final meeting at Hampden Park on Sunday.
Celtic extended their winning run to 11 games in all competitions thanks to veteran midfielder Brown and, although Accies have now gone nine without a win, boss Brian Rice will have been encouraged by his side’s spirited performance.
Neil Lennon had insisted he would field a strong side for the match despite the looming final.
In the event he made three changes, two in defence, with Moritz Bauer, Nir Bitton and Olivier Ntcham coming into the side at the expense of Jeremie Frimpong, Christopher Jullien and Tom Rogic.
Rice brought in Will Collar, Scott Martin and David Moyo, the latter for his first start since signing from St Albans in the summer, as he looked for a first win since beating Livingston in September.
It was a subdued start to the match before Christie’s opener.
Midfielder Callum McGregor slipped in Lewis Morgan and, when his angled drive from the left was parried by Accies keeper Luke Southwood, the Scotland international was first to react to drive the ball into the net for his 16th Hoops goal of the season.
To the irritation of their fans, the home side allowed Accies to garner some confidence.
Hoops keeper Fraser Forster cleared a Scott McMann cut-back with his foot before forward Mikel Miller blasted the ball over the bar.
In the 35th minute Parkhead winger James Forrest curled a shot from the edge of the box just past the far post then Christie and Ntcham missed the target with efforts.
Celtic continued their dominance after the break but Accies looked threatening in their sporadic counter-attacks and, in the 54th minute, Sam Stubbs’ header from a Miller free-kick appeared to be pushed onto the post by Forster.
Mikey Johnston replaced Ntcham on the hour mark as Lennon looked for more flair but it was the ever-dangerous Christie who sent a left-footed drive just over the bar after battling to retain possession, before defender Kristoffer Ajer headed a free-kick from Christie over the bar from close range.
Johnston then struck the post with a right-footed drive from 12 yards.
Forster had to make a save from Accies substitute George Oakley after another Hamilton break in the 88th minute and then Ogkmpoe stunned Parkhead when he equalised with a low drive from 10 yards.
However, Celtic piled forward in injury time and Celtic Park erupted when Brown sent the ball past Southwood.
Neil Lennon believes Celtic will be a different proposition in Sunday’s Betfred Cup final after it took a last-gasp goal from skipper Scott Brown to beat Hamilton 2-1 at Parkhead.
The Hoops boss did not enjoy what he watched as his side struggled despite taking the lead through Ryan Christie in the first half.
Accies substitute Marios Ogkmpoe stunned Celtic Park when he levelled in the last minute.
However, there was more drama to follow in the three added minutes when Brown drove into the Hamilton box and fired in the winner.
With Rangers letting slip a two-goal lead to draw 2-2 with Aberdeen at Pittodrie, Lennon’s side are now two points clear of the Light Blues at the top of the table ahead of their meeting at Hampden Park on Sunday.
“I didn’t enjoy it,” said the Northern Irishman, who revealed Odsonne Edouard and all the other players battling back from injury will be assessed on Thursday.
“Yesterday, you guys were all asking me about the cup final.
“There is no doubt subconsciously that would have played on the minds of some of them going into the game tonight, you could feel it around the stadium as well.
“We were flat at times and then the psychology towards the end when we looked a bit tired and that’s mental fatigue not physical.
“And then the psychology of the game again, they equalise and then we come to life again and get the goal from Scott.
“It is a huge goal, it keeps the momentum going. That’s 11 wins in a row, we are top of the league and we rested some players.
“You will see a different team on Sunday, mentally and physically, no question of that and that is just natural going in to a cup final.”
Accies boss Brian Rice believed referee Alan Muir missed a foul in the build-up to Celtic’ winner.
He said: “It’s an angry dressing because the boys feel we should have got something.
“Celtic are the champions and keep going until the 93rd minute but you saw a Hamilton team that were hungry.
“There are positives but I would rather have points and I don’t want to see pity in the dressing room.
“It might seem strange being disappointed but we deserved something from the game.
“A couple of decisions didn’t go our way either.
“The general consensus in the dressing was that it was a push from Leigh Griffiths on young Sam Stubbs.
“But I am not here to talk about referees, it’s done and dusted. Some go for you, some go against you.
“Did I speak to the ref? No. I have known Alan Muir for a long, long time but I don’t speak to him after games.”