Patrice Evra gave United an early lead, but Michu drew Swansea level before the half hour mark with his 13th goal of the campaign.
United dominated the second half and twice hit the crossbar, but Swansea defended brilliantly to pick up a share of the spoils.
After a bright start from Swansea it was United who created the first noteworthy effort of the game on 13 minutes when Wayne Rooney saw his swerving shot kept out by a smart save from Michel Vorm.
Vorm came to Swansea's rescue again a minute later when he produced a smart reaction save to keep out Ashley Young's shot inside the box after Carrick played the winger in on goal with a fine pass.
United made the breakthrough on 16 minutes with Evra heading home his third Premier League goal of the season after rising highest to meet Robin van Persie's whipped in corner with the ball going in off the far post past Leon Britton on the line.
Swansea came roaring back into the game and they drew level just before the hour mark with Michu netting his 13th goal of the season.
Wayne Routledge slipped in Jonathan de Guzman inside the box and his shot across goal could only be palmed out by David de Gea to the grateful Michu who tapped home the rebound from close-range.
United had a great chance to restore their lead just before the break when Evra got in down the left and pulled the back for the unmarked Rooney 12 yards from goal, but the England man could only fire his shot straight at Vorm.
Swansea started the brighter of the two sides at the start of the second half and De Gea was twice called into the action to keep out efforts from Kemy Agustien and Wayne Routledge.
Ashley Williams produced a superb block to deny Young inside the box after van Persie teed up the winger for a shot on goal.
The visitors were beginning to turn the screw and United were inches away from restoring their lead on 67 minutes when out of nothing van Persie saw his instinctive volley from Rooney's cross loop onto the crossbar.
Things threatened to boil over 16 minutes from time when Williams and van Persie were both booked after a melee involving the pair when after being fouled Williams drilled the ball into the Dutchman's head while he was on the ground.
United struck the cross bar for a second time on 78 minutes when Carrick got on the end of van Persie's corner and was unlucky to see his effort come back off the woodwork.
Swansea were defending in numbers and Williams and Davies produced vital blocks inside the box to deny van Persie and Young respectively as United piled on the pressure.
United threw everything at Swansea in the closing stages, but they could not find a way past as they had to settle for a point to move four points clear at the top of the table.