The Tannadice side looked to be heading for their first win in the east end of Glasgow for 21 years as they resolutely defended a lead given to them in the 38th minute by midfielder Stuart Armstrong.
That long-awaited win would have been almost certain had striker Brian Graham not missed an incredible sitter in the 65th minute when he somehow made a hash of a Nadir Ciftci cross, and the visitors were made to pay in the second minute of three added on as Mulgrew rose to flick a cross from Emilio Izaguirre into the net.
Celtic had been profligate all afternoon but the equaliser kept them unbeaten in the league after 11 games as they look ahead to their crunch Champions League clash with Ajax in Amsterdam on Wednesday night, where victory could strengthen hopes of qualifying through to the knockout stages.
Inverness moved back up to second in the table with a hard fought but deserved 2-1 victory over Kilmarnock.
Kilmarnock briefly threatened to take something out of this match when Darren Barr struck from close range in the 76th minute - but they could not find an equaliser.
Hearts returned to earth with a bump following the high of their League Cup derby success as goals from Nigel Hasselbaink and Stevie May earned St Johnstone a 2-0 win at Tynecastle.
Hearts entered the field to a warm reception following their Easter Road heroics, which were followed by the resignation of Hibs boss Pat Fenlon - but they were looking for a more positive consequence in the shape of a launchpad for a Scottish Premiership revival.
The home side made a bright start but Hasselbaink headed home in the 28th minute and fellow semi-finalists St Johnstone never looked like relinquishing their lead.
May netted his ninth goal of the season in the 53rd minute when he curled home brilliantly following a spell of intense pressure and, although Hearts kept pressing for a way back, Saints held firm to record a third consecutive victory and clean sheet and consolidate their place in the top six.
- Sports & Recreation