A rare error by Rangers' goalkeeper, Andy Goram, led to the 66th-minute winner, with the ball spinning off the head of Gough after Goram misjudged a cross from Andy McLaren.
Tommy McLean's United move up to fifth in the Premier Division as a result of the season's second league defeat for Rangers - but they lost their substitute Neil Duffy, who was carried off to hospital with concussion after a bad tackle by David Robertson.
United made a bright start as they forced Andy Goram into a save within the first 20 seconds, when Erik Pedersen set up a shooting chance for Lars Zetterlund. Robbie Winters was then gifted another chance when Craig Moore was caught in possession and the United striker's shot flashed wide of Goram's right-hand post.
United's goalkeeper Sieb Dykstra did his bit, saving United on several occasions in the first half. Just before half-time Alex Cleland worked a one-two with Ally McCoist - but he could not beat the keeper.
Rangers, and Paul Gascoigne in particular, were indebted to Goram just after half-time when he plunged to save a shot by Andy McLaren after Gascoigne was caught in possession. McLaren's next involvement was to collect a deserved yellow card for a foolish foul on Robertson, but it was from the winger's cross that United took the lead - thanks to Gough.
McLean said: "It was a good result for us and a great one for Scottish football. Before the match everybody was talking about how many points Rangers could go clear of Celtic - and not about how we could close the gap on them."
Dundee United: Dykstra, Bowman, Malpas, Pressley, Perry (Duffy, 65; McKinnon, 76), Pedersen, Olafsson, Zetterlund, Winters (McSwegan, 87), McInally, McLaren.
Rangers: Goram, Cleland, Robertson, Gough, Petric, Moore, McInnes (Steven, 71), Gascoigne, McCoist (Andersen, 71), Ferguson, Laudrup. Substitute not used: Wilson.
Referee: W Young (Clarkston).Reuse content