Captain Jon Daly's last-minute penalty came after the Terrors had raced into a 3-0 lead inside eight minutes - before Billy McKay's hat-trick followed by Gary Warren's 86th-minute header looked like earning the points for the second-placed visitors.
United went ahead through Josh Meeking's early own goal before further strikes from Daly and Rudi Skacel seemed to have them home and dry, until McKay sparked a comeback which few inside Tannadice would have predicted.
Motherwell produced a fantastic second-half comeback to defeat Hibernian 3-2 at Easter Road.
Eoin Doyle headed the hosts ahead five minutes before half-time and doubled his tally nine minutes after the break when he raced onto Paul Cairney's high ball and rocketed past Darren Randolph.
Jamie Murphy tapped in from close range to cut the hosts' lead and he glanced home a leveller with 10 minutes remaining.
Hibs looked shell-shocked after losing a commanding lead and things got worse for Pat Fenlon's side when substitute Robert McHugh converted Henrik Ojamaa's cross two minutes from time.
James McPake was dismissed for a second yellow card during injury-time to complete a miserable afternoon for the home side, who now trail Motherwell by three points in the Clydesdale Bank Premier League table.
Hearts took the lead twice but ultimately had to settle for a point in a thrilling 2-2 draw with St Johnstone.
The visitors opened the scoring through John Sutton and Andrew Driver gave them a 2-1 half-time advantage after Rowan Vine had levelled the scores.
Steve MacLean equalised after the break in a match filled with scoring opportunities and questionable defending.
Scotland midfielder Liam Kelly bagged a double for Kilmarnock as they scored a vital win over 10-man Aberdeen at Pittodrie.
Kelly opened the scoring from the penalty spot after Dons goalkeeper Jamie Langfield had been sent off for bringing down Borja Perez, and the midfielder struck from the edge of the area to double his side's lead in the dying stages.
Ross County came from a goal down to earn a draw against bottom club Dundee in an intriguing tussle at Global Energy Stadium.
Iain Davidson gave Dundee the lead eight minutes from the break, but County hit back strongly and substitute Gary Glen fired home a deserved equaliser with 16 minutes remaining.