Celtic showed their incredible fighting sprit and resilience to come back and score two late goals as Martin O'Neill's side saw off a spirited United challenge.
Alan Archibald had put the visitors into the lead in the 61st minute, but an equaliser by substitute Shaun Maloney was followed by a penalty by Chris Sutton as Celtic maintained their 100 per cent home record and extended their lead at the top to 13 points.
It was almost as though the news that Rangers had dropped a further two points earlier in the day at Aberdeen had taken the edge off Celtic's appetite as they turned in as lack-lustre a first half display as any this season.
When Alan Archibald touched home Barry Robson's curling free-kick, it finally produced a reaction from the home side who proceeded to lay siege to the United goal.
The visitors dropped deeper and deeper in defence and just when Celtic were becoming increasingly frustrated they made the breakthrough.
Sutton gained control near the byline before cutting the ball back to Maloney who held off Mark Wilson's challenge before sidefooting home.
If that goal was clear-cut, there was certainly controversy over Celtic's winner when Stilian Petrov went down under a challenge by Stuart Duff.
It looked a soft award and the United players, particularly Paul Gallacher, were incensed at Petrov's actions.
Sutton however was unmoved and nonchalantly stroked his spot-kick home past Gallacher and into the corner of the net.
This victory made it 23 league wins in a row for Celtic.
After the match O'Neill was a relieved man saying: "I'm delighted to have won.
"After the Rangers result it was vital that we got something out of the game though victory looked improbable after they scored.
"My side showed fantastic resilience. We kept going even though we didn't play well at all today.
"We were off colour, the pitch was heavy and maybe the Rangers result made us too relaxed.
"At half-time I doubted whether we had made three concerted efforts at the United goal although after Maloney's goal the floodgates opened." For visiting manager Ian McCall it was a hard result to take. "We'd have been disappointed at the end if we had only taken a point," he said.
"We were denied a stonewall penalty in the first half when Bobo Balde clearly handled." Regarding Petrov's penalty, McCall would only say: "It was cute how Petrov won the penalty - that's how Martin would describe it anyway.
"I have a lot of angry players in there and Duff claims he never touched him." O'Neill was certainly accurate when he described how off-colour Celtic were in the first half with only Stephen Pearson showing the required urgency.
United were content to sit back although they will point to the Balde incident as another valid penalty claim that has been turned down in Glasgow, following their game against Rangers before Christmas.
It was only when Archibald put United ahead that Celtic started to show some interest.
Gallacher did well to parry Stanislav Varga's close-range header although by that time Celtic could have been two down.
Rab Douglas in the Celtic goal recovered well to turn Jim McIntyre's shot for a corner after errors by Balde and Varga had allowed Billy Dodds and Charlie Miller to create the chance.
Henrik Larsson headed narrowly over before the United keeper saved well at his post from a stinging Petrov shot.
It looked as though it would not be Celtic's day when Duff cleared Maloney's effort off the line, after the sub had made an immediate impact when he came on for Pearson.
After Maloney's goal, and Sutton's penalty, there was further heartache for United when Dodds was sent off near the end for a second bookable offence.
The ex-Rangers striker had been booked along with Miller for dissent after the penalty and his tackle on Maloney left the referee little option than to order him off.
After Dodds' departure United threw more players up front than they had done all afternoon in a desperate but fruitless search for an equaliser.
By that time, Celtic were in control with sub Paul Lambert slowing play down at every opportunity and Larsson and Lennon showing their experience by keeping the ball down at the corner flag until the final whistle.
Man of the Match: Neil Lennon- Despite taking a bad knock in the first half, the flame haired Celtic midfield player drove on his team-mates at every opportunity.
He never hid despite things not going for his side and pushed forward near the end as his side pushed for the breakthrough which eventually came in the last ten minutes.