After Bruce Djite's 22nd-minute strike gave the Reds the early advantage, Harry Kewell coolly capitalised from the penalty spot in the 61st minute following an errant Nigel Boogaard tackle in the box.
Former Leicester City and Leeds United stalwart Patrick Kisnorbo scored a beauty 18 minutes later to give Heart the edge before Griffiths' last-gasp effort spared United's blushes.
The visitors started the clash confidently, buoyed by last week's drought-breaking win over the Newcastle Jets and Thursday's announcement of the club being acquired by English Premier League giant Manchester City.
After a wonderful Jeronimo Neumann cross, Awer Mabil's nod forced Heart gloveman Andrew Redmayne into a fine lunging stop to his left, but the parried deflection rebounded into the path of Djite, who headed it home past Redmayne's follow-up dive.
Mate Dugandzic's 42nd minute free kick just outside the box was denied by a sprawling Eugene Galekovic, as the Reds did enough to cling on to their 1-0 half-time advantage.
Carrusca did his best to give the home side the lead back but his close-range header in the 74th minute bobbled fractionally wide, Adelaide's frustration evident when McKain's brush with earned the former Reds skipper a yellow card and almost resulted in an all-in skirmish.
In contrast, Melbourne wouldn't be denied in the 79th when experienced centre back Patrick Kisnorbo made it 2-1, glancing home a header which eluded Galekovic on the back of an absolute peach of a Ramsay curling free kick outside the box.
The Reds looked gone before Griffiths, who substituted on in the 62nd minute, got on the end of Carrusca's corner, volleying it home to draw United level.
Kewell's chance for a brace went begging in extra time when he slewed a free kick wide before Argentine Jonatan Germano's 94th-minute booking added further drama to a gripping closing salvo.
Adelaide United 2 (Djite 22, Griffiths 90)
Melbourne Heart 2 (Kewell 61p, Kisnorbo 79)
