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.
Despite extending their undefeated run to seven on the trot, the Reds will be disappointed with having to share the spoils against Melbourne, vast underdogs, albeit with recent momentum.
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.
With captain Kewell and Jason Hoffman creating drive from the right-hand side, Melbourne racked up a couple of corners but could muster few good looks against a surprisingly hesitant Reds outfit.
But after their slowish start, United gradually began to dictate the tempo and created some Heart tremors in the 12th minute when Marcelo Carrusca stepped up for a free kick, only to push it well high, after being felled by yellow-carded Massimo Murdocca just outside the 18-yard box.
Adelaide were pushing hard and rewarded in the 22nd minute when Djite nodded.
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.
United were attacking with purpose and, piloted wonderfully by former skipper Jon McKain, defending with aplomb.
Ex-Adelaide midfielder Iain Ramsay chanced his arm with some dangerous forays and led a Heart revival late in the half with back-to-back corners.
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.
On the comeback trail from a broken leg, giant marquee Dutchman Orlando Engelaar still looks short of a gallop and substituted out of the contest in the 54th minute, while Maltese striker Michael Mifsud was subbed out seven minutes later after spraying a sitter.
Looking for answers, the visitors were reprieved when Nigel Boogaard was pinged for a reckless studs-up tackle deep in the box.
Kewell stepped up to the spot and slipped the penalty into the right-hand corner in the 61st minute to equalise.
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)
Crowd: 11,100 @ Coopers Stadium
