Goals from Harry Kewell and Patrick Kisnorbo looked to have given Heart a surprise win until Griffiths struck in the final minute to secure a draw for United.
After an even first 20 minutes, the home side took the lead when Bruce Djite reacted fastest after Awer Mabil had seen a header saved by the goalkeeper.
However, after winning their first game of the season last weekend, Melbourne looked confident, and they were back on level terms after an hour thanks to a penalty from Kewell.
Eleven minutes from time, Kisnorbo looked to have given Heart their second win in the space of eight days with a fine header from a free kick.
However, Adelaide pressed forward and got their reward in the final minute of the match when Griffiths slotted home his first goal for the club after a corner was delivered into the box by Marcelo Carrusca