Prince Buaben soon capitalised on a rare error from John Ruddy and Danny Swanson broke away midway through the second half to finish off Motherwell's uninspired comeback attempts.
Well had lost just three goals in their previous 12 SPL games as they went on an unbeaten run but they ended the match a dispirited side as they fell four points behind United and dropped to fifth.
Dundee United showed no sign of fatigue from Wednesday's Scottish Cup win over Rangers, although injured goalscorer David Robertson was among five players dropping out.
Motherwell had enjoyed a two-week interval after their previous match - which included a week off for the players - and they initially looked to have reaped the rewards as they made a bright start.
Jamie Murphy came close to connecting with crosses from Tom Hateley and Lukas Jutkiewicz the latter delivery being cleared by Keith Watson from inside his six-yard box.
The game entered a scrappy phase which culminated in a straight red card for Dixon.
Jim O'Brien had caught the full-back moments earlier with a raised boot without punishment from Willie Collum, although the Motherwell winger immediately put the ball out of play to allow his opponent to receive treatment.
The gesture failed to placate Dixon, who exacted retribution by thrusting his studs into O'Brien's thigh at the next opportunity.
The 10 men had the first chance after the dismissal when Goodwillie hooked the ball over the onrushing Ruddy but inches beyond the far post from a tight angle after Stephen Craigan's clearance was charged down.
The unmarked Murphy missed a glorious chance moments later when he headed O'Brien's cross wide from eight yards.
Well had the ball in the net in the 33rd minute when Keith Lasley's shot deflected in off John Sutton after a half-cleared free-kick, but the offside flag was immediately raised despite protestations from Craigan.
United took the lead when Webster headed down Craig Conway's corner and the ball edged over the line off the foot of Goodwillie.
The hosts doubled their advantage in the 44th minute when Ruddy, who has kept 15 SPL clean sheets this season, failed to control Craigan's passback and Buaben dispossessed him before rolling the ball home from 10 yards.
Murphy had a tap-in goal disallowed two minutes after the restart, this time a clear-cut decision with Jutkiewicz flagged offside as he initially tried to turn the ball home following a Stevie Hammell corner.
O'Brien soon volleyed high over from inside the box but the pressure dissipated with Motherwell's possession all coming in front of United's back line.
Craig Brown brought on Chris Humphrey but his team still lacked a creative edge and they fell further behind in the 72nd minute after United broke following a corner.
The away defence parted as Swanson strode forward and the midfielder placed the ball high into the net from 20 yards as Ruddy rushed out.
Michael McGlinchey and Giles Coke came on for Motherwell and both had several efforts at goal but Dusan Pernis was not seriously troubled.
The hosts lost the impressive Webster to a head injury but Darren Dods enjoyed a comfortable run-out in the final stages.