Both sides had chances from there. Joe Shaughnessy's header bounced just over for Dundee, while Sam Dalby could not steer his own headed effort on target.
But as United threw on a succession of attackers in search of a leveller, Dundee struck the telling blow late on.
Substitute Emmanuel Adegboyega - by then United's sole defender - was shown a second yellow card for desperately shoving Murray in the box.
The Dundee fan picked himself up to seal the win from the spot before gleefully running the length of the Tannadice pitch to celebrate in front of the United fans.
"That's the performance we're capable of," Docherty said. "I take a huge amount of pride, not just in the result.
"We've won here for the first time in 20 years. There was a lot on that today. I saw the players' spirit and mentality. I had confidence and trust in them.
"It was a performance from a team that displayed absolutely everything."
It is Dundee's first league win since 5 January and their first top-flight victory at Tannadice since a 2-1 triumph under Jim Duffy in August 2004.
Tony Docherty's side move to within one point of Kilmarnock in 10th and are five clear of bottom side St Johnstone.
United, meanwhile, stay fifth, three points ahead of seventh-placed Motherwell with three games left before the split.