It was a bright start from City as they earned a corner within the first minute, but Adelaide goalkeeper Eugene Galekovic was able to punch the ball clear.
As the first half went on, the game became more open and City came first to breaking the deadlock with some help from David Villa.
The Spaniard crossed to Damien Duff, whose effort bounced off the post after Galekovic almost let the ball slip past his line.
The opening goal came in the eighth minute for City when Patrick Kisnorbo unleashed a header into the far left corner of the net after connecting with Duff's corner kick.
Villa, who is due to join up with MLS franchise New York City in the coming weeks, began to dominate the attack and created a chance of his own when he aimed for goal with a left-footed drive, but the keeper was able to save and stop the ball from slotting into the bottom right corner.
Adelaide began building up the pressure in search of an equaliser and it eventually came when Tarek Elrich whipped in a thunderous cross into the box to find the head of Bruce Djite, who knocked the ball into the back of the net.
After the break, City came close to taking the lead with an attacking play, which resulted in the ball flashing across goal, but Aaron Mooy failed to get there in time for a tap-in.
City earned themselves two quick free kicks just outside of the area, but neither Villa nor Mooy could make the most of them as both aimed for goal and missed the target.
At the other end, Djite was brought down by Kisnorbo just yards outside of the box, but the Adelaide player and his teammates were furious at the referee for failing to issue a red card as they believed that the forward was clear on goal.
Villa displayed his quality by storming past a number of Adelaide shirts to get into the box and unleashed a shot at the near post, which was pushed away by the keeper, but the follow-up effort sky-rocketed.
Marcelo Carrusca was given the responsibility to take the penalty, and he eased it into the back of the net to extend Adelaide's unbeaten run to four games.