Celtic defeated Dundee United 0.
Despite being reduced to 10 men, Liel Abada scored a late goal to help Celtic narrow the gap on Rangers.
Rangers drew away to Ross County, giving Postecoglou's side the chance to pile on the pressure before the Old Firm derby.
The home side was denied by a series of saves by Benjamin Siegrist before he was sent off.
Abada scored from close range as the game entered added time.
It looked as if United would leave Celtic Park with their second draw of the season.
Celtic Park has been without a win for United in 46 meetings since December 1992.
Celtic were looking to pay a winning tribute to their former manager, who died this week, and had just heard Rangers had dropped a couple of points.
One of five changes made by United head coach Thomas Courts was the start of Adrian Sporle at left wing-back, and the Argentine flashed an early warning shot across the face of the Celtic goal.
The United rearguard was stubborn and brave in their denial of a breakthrough despite Celtic being as dominant in possession as expected.
When the home side did manage a sight of goal, Siegrist was in typically obstinate form and saved Abada and Matt O'Riley before the latter missed a great chance.
Jota and Josip Juranovic were thrown on at half-time in order to give them a rest before the derby.
Jota made an immediate impression, setting up O'Riley for a head flick that forced another Siegrist save before laying it on a plate for Giorgos Giakoumakis, only for the goalkeeper to divert the strikers shot.
When Bitton was shown a second yellow card after hauling down United substitute Declan Glass, it looked as if Celtic's frustrating afternoon was over - and there was still time for it to get worse.
Abada made no mistake from close range after Jota threw in another dangerous cross, sending Celtic Park into a mix of relief and rapturous applause.
Celtic now have the chance to go top if they defeat Rangers on their own patch in the Old Firm derby.
Jota proved again that his presence is key to their title challenge when he came on as a substitute for Ben Doak.
What a difference late goals for County and Abada have made.
Glasgow will leave Celtic Park with a lot of positives despite the fact that United failed to match their draw earlier in the season.
It was a frenetic finish. I am happy with the result. They found something within the players.
We weren't at our best in the first half. You think we have to find something special as long as the game continues. They have risen to every challenge for the last three and a half months.
The players gave absolutely everything and it is very cruel.
The boys executed the gameplan well. We were bitterly disappointed that we lost that late goal, which is testament to Celtic for pushing right to the end.
Both Celtic and United will play local derbies on Tuesday and Rangers will play Celtic on Thursday.
Formation 4-3-3.
56Ralston20Carter-Vickers4Starfelt5Scales11Abada7Giakoumakis49ForrestFormation 3-6-2
20Neilson18Butcher33McMann7Niskanen17Meekison19Levitt23Harkes3SpörleCeltic defeated Dundee United 0.
The second half ended with Celtic winning.
An attempt was missed. Ben Doak missed his shot from a difficult angle on the left. Greg Taylor assisted.
Foul by Liel Abada.
In the defensive half, Ilmari Niskanen won a free kick.
There is a yellow card for a bad foul shown to Declan Glass.
James McCarthy won a free kick.
Foul by Declan Glass.
The yellow card shows Liel Abada for excessive celebration.
Goal! Celtic beat Dundee United. Liel Abada shot from the centre of the box to the top left corner. Jota assisted by a cross.
An attempt was missed. Liel Abada's headbutt misses to the left. Jo Jursipanovic assisted with a cross.
Carter-Vickers won the free kick in the defensive half.
The foul was committed by Marc McNulty.
The attempt was blocked. The shot from the right side of the box is blocked by the box. Josip Juranovic assisted.
An attempt was missed. The left footed shot from the centre of the box misses to the left. Ben Doak assisted.
Celtic. Giorgos Giakoumakis was replaced by James McCarthy.
Carter-Vickers won the free kick in the defensive half.
Peter Pawlett of Dundee United had a foul.
The attempt was blocked. Ian Harkes' shot from outside the box is blocked.
The second yellow card was for a bad foul.