Motherwell struck after just two minutes when Steven Hammell fed James McFadden, and he skipped past John Rankin before laying the ball in front of Michael Higdon to lash home from 25 yards via the right post.
United almost levelled when youngster Ryan Gauld turned in acres of space before exchanging passes with Rankin and chipping the ball against the top of the crossbar.
Hammell also started the move for Motherwell's second when his throw-in came off Higdon's thigh to Henrik Ojamaa, and the little forward ran through the heart of the United defence and poked the ball past Rado Cierzniak.
Stuart Armstrong left Tom Hateley standing and swept the ball to the near post but Jon Daly's flick was blocked by the chest of follow Irishman Darren Randolph to deny United a route back into the game.
And Randolph produced an excellent save to claw away a Gary Mackay-Steven shot with Daly lurking behind the keeper with the goal at his mercy.
Higdon secured the points 10 minutes after the break when he took a pass from Chris Humphrey at the edge of the box and turned to dispatch a low drive into the far corner.
Willo Flood then hit the post with a curling free-kick from 25 yards with Randolph beaten at full stretch, before Daly (69) slipped the ball through the keeper's legs from Mackay-Steven's assist.
And Randolph ensured there would be no nervy finish for Motherwell when he got down to turn a powerful Mackay-Steven drive round the post 12 minutes from time.