Scottish Premiership Wrap: Inverness close gap as Celtic drop points
The visitors took the lead at Parkhead seven minutes before half-time, Stuart Armstrong collecting Nadir Ciftci's superb pass before lofting the ball over Fraser Forster.
Anthony Stokes and Amido Balde both wasted opportunities as Celtic looked to find a way back in to the match, but they should have fallen further behind just after the hour mark when United striker Brian Graham miscued a header from close range.
And Jackie McNamara's men were made to pay for that missed chance as Mulgrew nodded in the leveller two minutes into added time to seal a share of the spoils.
Celtic's slip-up allowed Inverness Caledonian Thistle to narrow the gap at the top, and they took full advantage with a 2-1 home win over Kilmarnock.
Winger Aaron Doran's deflected strike put the hosts in front after seven minutes, and their advantage was doubled early in the second half through defender Graeme Shinnie.
Darren Barr did pull one back for Kilmarnock 14 minutes from time, but they could not complete the comeback as Inverness moved five points adrift of Celtic.
Hearts remain bottom on minus seven points after they fell to a 2-0 home defeat against St Johnstone.
Dutch striker Nigel Hasselbaink opened the scoring for the visitors, who made sure of the points with a strike from midfielder Stevie May eight minutes after the interval.
St Mirren were undone by the weather as their clash with Ross County was abandoned with the hosts leading 2-0.
Goals from midfielder Paul McGowan and forward Steven Thompson had put St Mirren in command.
However, referee Willie Collum called the players off the pitch in the second half due to a waterlogged pitch, which was eventually deemed unplayable despite the best efforts of the ground staff at St Mirren Park.