Dundee United 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.
Inverness moved back up to second in the Scottish Premiership table with a hard fought but deserved 2-1 victory over Kilmarnock.
Aaron Doran's deflected effort in the seventh minute put Inverness in front at the break, before Graeme Shinnie moved them further in front with a sublime finish in the 52nd minute.
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.
Ross County experienced a second abandoned match in nine days as a waterlogged pitch halted their Scottish Premiership fixture at St Mirren Park six minutes into the second half.
The home side were two goals up against their Dingwall opponents when referee Willie Collum decided the game could not continue.