Murray headed home Finlay Robertson's in-swinging corner to give the hosts the perfect start - and that set the tone for the rest of the first half.
The hosts were ferocious in their press, strong in the tackle and bright in attack, with Murray, Lyall Cameron and Oluwaseun Adewumi all heavily involved.
Murray curled just over the top and had a close-range shot well blocked by Emmanuel Adegboyega, while Aaron Donnelly headed a Robertson corner just over as the home side probed for a second goal.
United twice had goals disallowed in the opening period - Sam Dalby turned home after Will Ferry's cross drifted out of play and Vicko Sevelj fired home from the edge of the box after referee Matthew MacDermid had blown for a foul in the middle.
The game became more frantic after the break, with all the same intensity of the first.
Cameron had a shot just over the top for the hosts, before United's Louis Moult headed Ferry's excellent cross wide and Kevin Holt shot straight at Dundee goalkeeper Trevor Carson.
Jim Goodwin's side turned the screw in search of a late equaliser, but Port Vale loanee Ruari Paton was denied a debut goal by an outstanding Ethan Ingram block, while Carson denied Declan Gallagher and Ferry with excellent saves.
Substitute Kristijan Trapanovski thought he had found a way through in injury time, but his curling effort bounced clear off the post, and Dundee hung on to claim their spot in the next round.