A dominant first-half display proved enough for Graham Alexander's side to claim their second win in succession after the Steelmen had gone three months without one.
United made a game of it in an absorbing second half and Ryan Edwards headed home in the 81st minute.
But Motherwell held on to move six points above bottom side Hamilton and three behind seventh-placed United, who have now not won in eight matches.
Alexander made seven signings during the transfer window but none of them started.
He was forced into one change following last week's victory over Ross County with skipper Declan Gallagher missing with a muscle injury. Ricki Lamie came in and Stephen O'Donnell, fresh from signing another short-term deal on Monday, wore the armband.
United switched tactics after conceding seven goals in home defeats against St Mirren and Hibernian. Micky Mellon moved to a back three with Calum Butcher dropping back and Edwards coming in to replace the benched Mark Reynolds and absent Mark Connolly.
Mellon made four changes in all after admitting trying to get some attacking width into their play did not work, but his side endured a torrid time after an uneventful first 15 minutes.
Benjamin Siegrist kept United level with a good tip-over from a Bevis Mugabi header before Tony Watt squandered two excellent chances.
The striker side-footed wide from six yards when he had more time and space than he appeared to realise, and then failed to connect with a high boot when he might have nodded home Allan Campbell's ball across the face of goal.
Watt was soon involved in the 23rd-minute opener, heading on Barry Maguire's corner for Cole to turn and nudge the ball over the line.
Motherwell doubled their lead when a fine one-touch move ended with Long firing a powerful strike into the corner of the net from 22 yards following Cole's lay-off.
The goalscorers both had other efforts off target as Motherwell found space to run at United's defence.
Mellon switched to a 4-1-4-1 formation for the second half and his side emerged with attacking intent. They had three loud penalty claims rejected by Don Robertson inside seven minutes with Marc McNulty, twice, and substitute Peter Pawlett going down.
The game had completely opened up with United throwing men forward and the hosts creating chances on the break.
Siegrist saved brilliantly from Watt and Cole, while Long came close, with Shankland flashing a shot just wide at the other end.
The game settled down slightly before United pulled one back from a corner which came after Liam Kelly had touched wide Butcher's 35-yard strike. Adrian Sporle delivered the set-piece and Edwards headed home powerfully at the near post.
Motherwell had their own penalty claim rejected when Jeando Fuchs stopped substitute Nathan McGinley heading an inviting cross with a high boot. The bloody-nosed McGinley had to go back off for debutant Sam Foley.
United immediately had an excellent chance when Sporle burst through but Kelly gathered his shot at the second attempt.
Siegrist saved from substitute Harry Smith as the hosts countered and they survived a goalmouth scramble to claim a crucial win.