John Rankin's 22nd-minute goal was cancelled out eight minutes into the second half by Aberdeen striker McGinn, who netted in Northern Ireland's midweek draw in Portugal on the occasion of Cristiano Ronaldo's 100th cap.
After 88 minutes, United should have sealed victory but Rankin fired wide across goal with a wasteful ball when he should have shot himself or passed to Johnny Russell.
While the Dons are undefeated since the opening day of the season, United remain at the wrong end of the Scottish Premier League, but played a full part in an entertaining New Firm derby which suggests their position will improve before too long.
United's most recent league win came in the Dundee derby on August 19 and they had to be patient for their first match in October following the postponement at Motherwell and the international break.
They were made to wait an additional 20 minutes as kick-off was delayed due to a road closure and there was a circumspect start.
Peter Houston's side had not scored in four SPL games, but captain Jon Daly, usually a striker, was selected at centre-back, a position he has played in numerous times previously, after United leaked seven goals in their last two games.
Houston, who endured a miserable international break as Scotland assistant manager, favoured a fleet-footed attacking trio of Russell, Michael Gardyne and Stuart Armstrong.
The Dons began confidently without testing Radoslaw Cierzniak in the United goal and it was Jamie Langfield who was first called into action.
Armstrong picked the ball up in the left channel, cut inside and fired a swerving right-foot shot which Langfield could not hold.
Mark Millar, twice, and Gardyne went close from long range soon afterwards before Daly hesitated, allowing his opposite number Scott Vernon to lay the ball off for Gavin Rae, who missed the target.
Russell was proving a handful and his run set up the opening goal.
He beat Mark Reynolds on the left before firing a shot Langfield could only parry back into the centre of the goal from where Rankin found the bottom corner with a scuffed shot.
Armstrong was also lively and he skipped by Isaac Osbourne on the byline before crossing along the six-yard line, but behind Russell.
Aberdeen came within inches of an equaliser when McGinn, buoyed by his strike in Porto, pulled the ball back for Vernon, whose snap-shot was turned around the post by Cierzniak.
A well-timed Russell Anderson tackle denied Russell before the Dons captain was left frustrated at the other end, his header from a corner blocked on the post by Millar.
United had four corners in quick succession early in the second half, two of them delayed when objects were thrown from the away end on to the pitch.
Ahead of the fourth, referee Iain Brines spoke to a policeman to report the incident, before Fraser's set-piece was headed on to the back post where McGinn swept the ball in.
Five minutes later a Willo Flood cross reached Watson inside the six-yard box, but Langfield blocked.
Armstrong shot narrowly wide as United continued to attack despite a poorly-timed tackle on Rankin which saw Vernon booked retrospectively.
Vernon then showed the more accomplished side of his game, finding Stephen Hughes, whose effort was blocked by Cierzniak.
Once again it was Russell who tried to respond for the hosts, running at the Dons defence before drilling an effort from 20 yards which Langfield saved well.
Millar missed the target with a free-kick and McGinn tried to find an opening for the winner at the other end.
Rankin's late indecision proved costly as United's wait for a win continued.