United is just four points ahead of Manchester City after Patrice Evra's headed opener was cancelled out by Swansea midfielder Michu.
Tempers flared late on when Swansea captain Ashley Williams cleared the ball straight into Robin van Persie's head from close range while the United striker was on the ground.
Van Persie reacted angrily and squared up to Williams with referee Michael Oliver booking both players.
United manager Alex Ferguson called it a "disgraceful act" by Williams and was furious tougher action was not taken by the referee.
"The Van Persie situation you can clearly see that he could have been killed," Ferguson said.
The Scottish manager wants the English Football Association to investigate the incident despite Williams already being sanctioned during the game.
"He should be banned for a long time because that was the most dangerous thing I've seen on a football field for many years. It was absolutely deliberate," Ferguson said. "The whistle has gone, the game has stopped and he has done that right in front of the referee, he could have killed the lad."
But Williams sought to downplay the incident.
"I tried to clear the ball, it hit him on the head," he said. "I was trying to clear the ball. He was obviously a bit angry about that. It was a much ado about nothing."
Swansea, which is 11th, started strongly with Michu slicing wide early on and forcing United goalkeeper David De Gea into making a save.
But United halted the host's building momentum as it took the lead.
Wayne Rooney had already tested goalkeeper Michel Vorm with a stinging drive before Evra netted in the 16th minute, arriving to glance a header in off the far post from a corner from Van Persie.
But Swansea recovered well from the blow and Jonathan de Guzman had the ball in the net two minutes later, but the effort was ruled out for Michu blocking off Jonny Evans as the midfielder advanced.
The Liberty Stadium fans were further angered when Phil Jones was given only a yellow card for a clumsy lunge on Ben Davies, although replays suggested referee Michael Oliver got the decision right.
But Michu levelled in the 29th minute. Evra was caught out by De Guzman's clever run and when De Gea could only parry the Canadian's angled shot, Michu was on hand to tap in.
Ferguson was visibly angered that Van Persie was not awarded a free kick after he and Chico Flores went to ground following the Spaniard's challenge seconds before the equalizer, and the United manager made his feelings known to the referee at the break.
The goal added an extra bite to Swansea's play and they had the better of the remainder of the opening half.
De Guzman and Wayne Routledge were becoming increasingly influential for the hosts and the former should have done better after working his way into the United box on the byline, only to shoot straight at De Gea with several teammates well placed.
But United offered a reminder of its quality in first-half stoppage time when Evra easily got past Nathan Dyer and intelligently pulled the ball back to Rooney, but the England forward fired directly at Vorm.
The second half started at a furious pace, with Michu heading over and Routledge's close-range strike stopped.
At the other end Williams made two superb blocks to stop goalbound efforts from Rooney and Ashley Young after Swansea twice surrendered possession cheaply.
United was finally exerting some authority and Van Persie came close to putting them back in front as he cleverly diverted a Rooney cross goalward, only to see his effort cannon away off the crossbar.
United substitute Javier Hernandez wasted another presentable chance as he failed to control Van Persie's astute pass and Vorm gratefully held the Mexican's deflected strike.
Worse was to follow for United when Rooney dispossessed Chico Flores but allowed the defender to get back and intercept a slack pass which should have given Hernandez an easy finish, much to the fury of his manager.
Swansea was by now under incessant pressure.
Vorm tipped Carrick's header against the crossbar while Van Persie and Young were denied as United couldn't find a winner to respond to champion City beating Reading 1-0 on Sunday.Suggest a correction