Two goals from Ryan Gauld and one each for Paul Paton and Andy Robertson saw the visitors to victory but they were helped considerably by poor defending from a makeshift home rearguard.
Motherwell, looking to extend their winning home run to six consecutive games, had to do so with a three changes in their back four with the ever-present of trio of Simon Ramsden, Stevie Hammell and Shaun Hutchison missing through injuries and suspension.
United will have come into the encounter full of confidence having won both of last season's encounters at Fir Park and not conceded a goal in their last four visits.
The home side had a couple of inviting crosses from Iain Vigurs and Lionel Ainsworth just elude John Sutton in front of goal before United took the lead after 15 minutes.
Adam Cummins barged into the back of Gauld to earn a booking. From the resulting free-kick, Stephen McManus sent the ball high towards his own goal and although Nielsen outstretched Nadir Ciftci to punch clear it went only as far as Gauld volleyed the ball into the empty net.
Before the home side could react they found themselves two behind. Zaine Francis-Angol set off towards his own goal line with Gary Mackay-Steven and Gauld on his heels and when the Well full-back lost possession, Gauld's cross found Ciftci.
Although his attempt was blocked, Paton pounced from eight yards to steer his header through a ruck of players to put the visitors in a commanding position with less than a quarter of the game played.
The Well defence looked dazed every time United headed towards the home goal and Ciftci almost made it three when he dragged his shot across the goalkeeper and just outside his post.
With 33 minutes played United appeared to have made it 3-0 when Gauld's pass found Robertson unmarked and the full-back slid the ball through the keeper's legs but his celebrations were cut short as the assistant referee flagged for offside.
Nine minutes from half-time Well came within inches of pulling one back but Sutton's angled drive was palmed across goal and behind by Radoslaw Cierzniak.
And on the stroke of half-time United came equally close when Ciftci's 25-yard drive skidded narrowly past Nielsen's right-hand post.
Motherwell started the second half with James McFadden replacing defender Cummins, but it was United who came even closer to extending their lead after 55 minutes.
Robertson's drive from the edge of the box took a wicked deflection and Nielsen did well to palm the ball onto the bar with the rebound bouncing just beyond the waiting Armstrong.
The goal was only delayed eight minutes. With Well down to ten men, and Euan Murray waiting to replace the injured Stuart Carswell, Gauld made the most of the space on the right side of the box to drill beyond the keeper and into the corner of his net.
Nielsen made a couple of great saves to deny United substitutes Brian Graham and David Goodwillie before conceding a fourth in injury time.
And it was Robertson who finally found the net at the third time of asking, hammering the ball high past the Well 'keeper to round off the scoring.
- Sports & Recreation
- Ryan Gauld