City had started the new campaign with a 1-0 win at Brisbane Roar but Sunday's game turned on the dismissal of Scott Jamieson in the 65th minute for two yellow cards.
A poor challenge on Yaya Dukuly saw the left-back given his marching orders and from the subsequent free-kick Louis D'Arrigo pulled the ball back for Tomi Juric to stroke home.
Captain Stefan Mauk doubled Adelaide's lead in the 76th minute as it claimed its first triumph of the 2020-2021 season.
Meanwhile, Central Coast Mariners has not won more than six games in a season in each of the past six campaigns but now have two wins from two this year after a 2-0 victory at newcomers Macarthur.
Expansion team Macarthur made a winning start to life at Western Sydney Wanderers, yet goals from Daniel De Silva and Jordan Smylie brought them back down to earth this time around.