Following up from last Wednesday's 3-0 Scottish Cup win, the Steelmen managed to edged ahead of the Terrors despite having just three shots on goal to the visitor's nine.
Stuart McCall's men took the lead against the run of play in the 26th minute when Jamie Murphy pounced on a defensive error, advanced toward the byline and tried his luck from a tight angle which proved successful for his 11th of the season.
Dundee were searching for their seventh-straight league win and were making all the running but it was the home side who went two ahead just before half-time. Chris Humphrey ran at the United defence after taking possession about 40 yards out and then un leashing a shot which deflected off Sean Dillon and spun over United keeper Dusan Pernis to make it 2-0.
Peter Houston's charges hit back with just over ten minutes remaining when Johnny Russell scored with a glancing header from a Paul Dixon corner and they then almost equalised in injury time but for 'Well gloveman Darren Randolph's top-shelf save from a David Goodwillie effort.