Liel Abada scored Celtic's winning goalCeltic narrow the gap on leaders Rangers to two points through Liel Abada's late goal against Dundee United after being reduced to 10 men.
Rangers' earlier draw away to Ross County handed Ange Postecoglou's side the opportunity to pile on the pressure ahead of Wednesday's Glasgow derby.
Goalkeeper Benjamin Siegrist denied the home side with a series of saves before Nir Bitton's 81st-minute red card.
But Abada fired home from close range as the game entered stoppage time.
Until then, it looked like United were about to do enough to leave Celtic Park with their second draw of the season.
Instead, Postecoglou's side remain unbeaten in 13 games, winning 12 of them, and leave United without a win at Celtic Park in 46 meetings since December 1992.
Considering they were looking to pay a winning tribute to title-winning late former manager Wim Jansen, and had just heard that leaders Rangers had dropped a couple of points in Dingwall, Celtic's first-half performance was strangely subdued.
Adrian Sporle's start at left wing-back was one of five changes made by United head coach Thomas Courts and the Argentine flashed an early warning shot across the face of the Celtic goal.
However, Celtic were dominant is possession, as you might expect, but found the United rearguard, in which 18-year-old Lewis Neilson was drafted in because of injuries, stubborn and brave in their denial of a breakthrough goal.
When the home side did manage a sight at goal, Siegrist was in typically obstinate form and saves from Liel Abada and Matt O'Riley came before the latter squandered a great chance when the unmarked recent signing from MK Dons fired wide from 12 yards.
Postecoglou was probably hoping to give Jota and Josip Juranovic a rest ahead of the midweek derby, but the duo were thrown on at half-time.
Jota made an immediate impression, the on-loan Benfica winger 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 somehow divert the striker's shot over the bar from point-blank range.
When Bitton was shown a second yellow card after hauling down United substitute Declan Glass, it looked like Celtic's frustrating afternoon was complete - and there was still time for it to become worse still.
Jota, though, threw in another dangerous cross and, when United failed to clear, Abada made no mistake from close range for his 12th goal of the season to send Celtic Park into a mix of relief and rapturous applause.
Man of the match - Benjamin Siegrist
United goalkeeper Benjamin Siegrist had a series of saves to deny Celtic until stoppage timeMore to follow.
Line-ups
Celtic
Formation 4-3-3
15Hart
56Ralston20Carter-Vickers4Starfelt5Scales
33O'Riley6Bitton41Hatate
11Abada7Giakoumakis49Forrest
15Hart56RalstonSubstituted forDoakat 68'minutes20Carter-Vickers4Starfelt5ScalesSubstituted forJuranovicat 45'minutes33O'Riley6BittonBooked at 81mins41HatateSubstituted forTaylorat 68'minutes11AbadaBooked at 90mins7GiakoumakisSubstituted forMcCarthyat 83'minutes49ForrestSubstituted forNeves Filipeat 45'minutesSubstitutes
2Jullien3Taylor12Soro16McCarthy17Neves Filipe29Bain57Welsh84Doak88JuranovicDundee Utd
Formation 3-5-2
1Siegrist
20Neilson18Butcher33McMann
7Niskanen17Meekison19Levitt23Harkes3Spörle
10Clark32Watt
1Siegrist20Neilson18ButcherBooked at 75mins33McMannSubstituted forGrahamat 50'minutes7Niskanen17MeekisonSubstituted forGlassat 79'minutesBooked at 90mins19Levitt23Harkes3Spörle10ClarkSubstituted forMcNultyat 62'minutes32WattSubstituted forPawlettat 79'minutesSubstitutes
8Pawlett9McNulty13Eriksson21Glass22Freeman26Mochrie27Appéré29Graham58Anim CudjoeLive Text
Match ends, Celtic 1, Dundee United 0.
Second Half ends, Celtic 1, Dundee United 0.
Attempt missed. Ben Doak (Celtic) right footed shot from a difficult angle on the left misses to the right. Assisted by Greg Taylor.
Foul by Liel Abada (Celtic).
Ilmari Niskanen (Dundee United) wins a free kick in the defensive half.
Declan Glass (Dundee United) is shown the yellow card for a bad foul.
James McCarthy (Celtic) wins a free kick in the defensive half.
Foul by Declan Glass (Dundee United).
Liel Abada (Celtic) is shown the yellow card for excessive celebration.
Goal! Celtic 1, Dundee United 0. Liel Abada (Celtic) right footed shot from the centre of the box to the top left corner. Assisted by Jota with a cross.
Attempt missed. Liel Abada (Celtic) header from very close range misses to the left. Assisted by Josip Juranovic with a cross.
Cameron Carter-Vickers (Celtic) wins a free kick in the defensive half.
Foul by Marc McNulty (Dundee United).
Attempt blocked. Matt O'Riley (Celtic) right footed shot from the right side of the box is blocked. Assisted by Josip Juranovic.
Attempt missed. Josip Juranovic (Celtic) left footed shot from the centre of the box misses to the left. Assisted by Ben Doak.
Substitution, Celtic. James McCarthy replaces Giorgos Giakoumakis.
Cameron Carter-Vickers (Celtic) wins a free kick in the defensive half.
Foul by Peter Pawlett (Dundee United).
Attempt blocked. Ian Harkes (Dundee United) right footed shot from outside the box is blocked.
Second yellow card to Nir Bitton (Celtic) for a bad foul.
