The home side, still with Kenny Black in caretaker charge following Stuart McCall’s departure at the start of the month, took a deserved early lead through Henrik Ojamaa’s excellent 20-yard strike.
Motherwell may be 10th in the Scottish Premiership table, but their last win in the league came against the Tangerines, and a similar result looked to be on the cards as they held the visitors at bay comfortably for the opening 30 minutes.
But United gradually worked themselves into proceedings, testing the home defence with increasing regularity as the match opened up.
An inch-perfect sliding tackle from Stephen McManus prevented Gary Mackay-Steven from getting his shot away for United, while Stuart Armstrong should have done better when put clear.
Motherwell still offered a threat of their own in attack, with Mark O’Brien’s header and Josh Law’s strike both drawing decent saves, while Ryan Dow’s effort was just stopped by Well keeper Dan Twardzik.
The second half continued in a similar vein with Callum Butcher, Aidan Connolly, Dow and Armstrong all going close before Motherwell’s dogged resistance came to an end.
It took United – flying high in third place in the table – until the 66th minute to find a way through, with Souttar nodding in Armstrong’s corner from close range.
The goal was a cue for United to increase their domination and it was no surprise when Watson – originally named on the bench but promoted to the starting line-up after Jaroslaw Fojut was injured in the warm-up – grabbed the winner with another header on 82 minutes, as he got on the end of Conor Townsend’s cross.
A last-gasp rally from Motherwell almost yielded an equaliser but Watson did enough to put John Sutton off as his injury-time header skied over the bar.