United seemed on course for the first Scottish Premiership derby win of the campaign when Blair Spittal netted twice in three minutes on Tuesday night, but Greg Stewart and James McPake rescued an unlikely point for the visitors.
Dundee created a great opportunity after 10 minutes when Paul McGinn drifted in behind Paul Dixon to smash a low drive which Luis Maria Zwick could only parry, and Nick Ross then blasted the rebound straight at the United goalkeeper.
Two minutes later, John Rankin tried his luck from 30 yards, and although the shot was on target, Dundee goalkeeper Scott Bain saved comfortably.
But Dundee were playing with confidence, and Kane Hemmings slipped a pass to Rory Loy before the striker weaved between two United defenders and crashed a low drive from just outside the box which Zwick got down to hold.
Zwick then had to be alert to clutch a powerful drive from McPake shortly before McGinn’s low delivery across the penalty area curled inches wide of the far post.
Hemmings timed his run to perfection to latch on to a weighted pass from Kevin Holt three minutes before the break and Loy was inches away from connecting with the driven cross.
Dundee came close again when Stewart easily turned his marker at a throw-in and forced Zwick to push his dipping half-volley away before Loy’s low return was diverted wide of the far post by the goalkeeper’s fingers.
United looked livelier after the break and Bain was forced into a diving safe to push away a crisp Rankin drive from 35 yards.
And 90 seconds later, Dixon sent a deep cross from the United left flank and Aidan Connolly tested Bain with a volley from the edge of the penalty area.
The breakthrough came on 64 minutes when Darko Bodul kept the ball in play from a deep cross from Dixon and laid the ball inside to Rankin, who feigned a shot to find Spittal inside the D, and the youngster skipped past McPake with a terrific piece of skill before dispatching a low drive into the corner.
With the United fans still celebrating, the home side doubled their advantage when substitute Chris Erskine played a measured pass through the Dundee defence and Spittal slipped the ball past the advancing Bain.
Spittal was denied a hat-trick when Bain got across to turn his free-kick round the post on 77 minutes.
But Dundee were given a lifeline nine minutes from time as Stewart exchanged passes with Paul McGowan on the corner of the penalty area and curled a wonderful shot into the far corner of the net.
Spittal again produced some great footwork to weave into the box and shot narrowly wide of the far post.
In a dramatic finale, United almost scored again in the third minute of stoppage-time when Bain tuned a shot from substitute Simon Murray against the underside of the crossbar.