City kicked its title defence off with a 2-1 win over Brisbane Roar in its opening match and looked set to make it two wins from two after goals from Jamie Maclaren and Andrew Nabbout gave it a 2-0 advantage after 61 minutes.
However, Adelaide struck twice in the closing stages, with Stefan Mauk halving the deficit before Ben Halloran grabbed a dramatic 90th-minute equaliser to stun City.
Elsewhere on Saturday, Sydney FC, which finished just two points behind City in second last season, slipped to a 1-0 defeat against Macarthur at home after an early Lachlan Rose strike.
Macarthur is now level on points with table-topping City and Wellington Phoenix, which beat Central Coast Mariners 2-1, but is behind the pair on goals scored and sits third.
Gary Hooper scored for the second game running for Phoenix in its home clash with the Mariners, who pulled one back through Marco Urena before Jaushua Sotirio fired in the winner.
The Nix is now unbeaten in 13 matches stretching back into last season and faces Western Sydney Wanderers in its next game.