The hosts saw Ryan Strain sent off 16 minutes from time to dampen hopes of a late rally, with Carl Veart's men now winless in four.
Adelaide is now six points behind City and has played a game more, with a trip to Melbourne to face its rival next on the agenda.
Having been held to a goalless draw last time out against Western United, Adelaide started brightly and Stefan Mauk should have capitalised on Josh Cavallo's fine delivery, but instead failed to control the ball when well placed.
Tomi Juric saw a strong header saved smartly by Oliver Sail as the home side pressed, but after a blank first half ambition had to take a back seat when Strain got his marching orders for a second yellow card after he kicked the ball away.
