The Giallorossi were staring defeat in the face with five minutes remaining, only for Roger Ibanez to head home from a corner to cut the deficit - before Tammy Abraham lashed home from close range in injury time to equalise.
Milan, the reigning Serie A champions, had taken an early lead through Pierre Kalulu, before substitute Tommaso Pobega doubled the lead with 20 minutes remaining.
At that point the game appeared over but, with time winding down, Roma sparked into life.
Ibanez's equaliser was markedly similar to Kalulu's opener - the Brazilian meeting a corner and steering his header home - and set-piece delivery proved crucial soon after, as Pellegrini swung in a late free-kick.
Substitute Nemanja Matic's initial header was well saved but Abraham reacted quickest to sweep home the rebound and spark wild celebrations from the away bench.
The Giallorossi are back in action on Thursday, as they host Genoa in the Coppa Italia. Their next league engagement sees Fiorentina come to the Stadio Olimpico next Sunday.
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