Danny Wilson's side were relying on Wycombe beating cross-city rivals Sheffield Wednesday, but the Owls' 2-0 win against Wycombe meant United finished in third and will face Stevenage in the two-legged semi-final.
Exeter were the better of the two teams in the first half and went in front through Alan Gow's sublime volley from the corner of the penalty box.
United's James Beattie was then sent off for a reckless two-footed lunge on Exeter's James Dunne but, if anything, it helped to galvanise Wilson's side.
Lee Williamson brought United level a minute before half-time when he swept the ball home from Matthew Lowton's cross and they went in from two minutes into the second period when Kevin McDonald's long-range shot took a huge deflection and found the net with Exeter goalkeeper Artur Krysiak wrong-footed.
Exeter were disappointing after the break, but they managed an equaliser deep into stoppage time when substitute Scot Bennett stooped to head home after Guillem Bauza's effort had been saved.