Winger Fabio Ferreira was the last-ditch hero for the 10-man Reds, scoring in the dying moments to help United climb their way out of a 2-1 hole and into third position on the ladder.
Ferreira produced the miracle but it was Marcelo Carrusca who struck the initial blow, with the the fifth-fastest goal in A-League history - just 26 seconds into the contest - to send the home fans into delirium.
But things progressively disintegrated for the Reds, whose top-two prospects looked gone before Ferreira's stunning equaliser.
David Williams slotted the first penalty before being denied with his second, but Mate Dugandzic's 42nd-minute strike gave Melbourne the lead at half-time.
The Heart moniker was apt considering the way they fought back from a nightmare start in which Carrusca, evoking memories of former Red Louis Brain's 11-second goal almost a decade ago, finished with a neat left-footer after a fine cross from Elrich, the visitors yet to touch the pill.
Jeronimo Neumann, substituted on for Marrone, came agonisingly close to making it 2-0 when he mopped up following Melbourne's defensive mix-up and cracked a volley into the right upright.
Moments later, Elrich's night was over, the defender handed his second yellow in as many minutes for making head contact on Melbourne left back Aziz Behich.
Galekovic pulled off another fine stop to deny a Mauk-Behich one-two in the 56th, before Melbourne goalkeeper Tando Velaphi showed similarly great athleticism and anticipation in denying Jeronimo's swerving attempt two minutes later.
They very nearly tied proceedings in the 76th minute when Jordan Elsey's flying nod crashed into the crossbar, Awer Mabil's follow-up reflex collected by Velaphi before Jeronimo's angled shot in the 86th missed by a coat of varnish.
Adelaide United 2 (Carrusca 1, Ferreira 90+2)
Melbourne Heart 2 (Williams 33pen, Dudzantic 42)
