Galekovic had to produce a number of excellent saves to deny the home side on many occasions throughout the contest.
Despite producing little in front of goal throughout the match, a swift attack late on, that was finished expertly by Mabil, sealed a smash-and-grab win for the Reds.
Adelaide had to do without its influential playmaker Marcelo Carrusca out injured and suspended Socceroos defender Tarek Elrich, while Sydney was boosted by the return from international duty of its top-scorer Marc Janko.
The Sky Blues appeared the more threatening early on with Bernie Ibini's speed on the right providing problems for the Reds defence.
On 17 minutes his dangerous cross was met by a sliding Alex Brosque in the box, with his effort saved by Galekovic and desperately cleared away by the visitor.
Sydney created another good opening five minutes later. This time the cross came in from the left and fell invitingly for Janko, who was denied at close range by the reactions of the Adelaide shotstopper.
The host's best chance of breaking the deadlock came just before half-time.
Neat interplay on the edge of the area left Brosque in the clear. He shot low and hard to the near post, but once more Galekovic pulled off an excellent save to turn the ball behind for a corner.
Adelaide's first shot on target came early in the second half as Bruce Djite was first to a low corner and steered the ball towards goal. It was comfortably dealt with by Vedran Janjetovic.
At the other end Sydney believed it had taken the lead when a Milos Dimitrijevic's free kick was turned in by Janko, who began celebrating with the home fans behind the goal.
But the assistant's flag was correctly raised and the goal was disallowed for offside.
Shortly after it was Galekvoic who once again came to the visitor's rescue. Adelaide failed to clear an Ibini cross which found Brosque free on the far post. He fired his shot at the near post, yet was once again denied by the Reds goalkeeper.
The same duo went head-to-head again on 56 minutes. Galkeovic's impressive reflexes this time denied a Brosque volley from just outside the six-yard box.
Somewhat against the run of play Adelaide struck on 82 minutes. A long ball from Galekovic was picked up by substitue Awer Mabil on the right outside the area.
He drove at the Sydney defence and cut inside before powering a shot inside the far post to make it 1-0 for the Reds.
Another fantastic Galekovic save in the dying minutes, after the Sky Blues had broken through, secured an unlikely three points for Adelaide.
Josep Gombau's side jumps to top place in the A-League, one point ahead of Wellington Phoenix and two in front of Melbourne Victory, Perth Glory and Sydney.
Sky Blues coach Graham Arnold praised his charges but criticised Adelaide's aggressive play that earned it seven yellow cards.
"I couldn't be prouder of the performance," he said.
"The result is what it is but our style of football and the way we're playing was outstanding.
"They gave everything they had and Eugene Galekovic had the game of his life.
"The most frustrating thing is every time you won the ball and you tried to break in transition they gave fouls away.
"It's not great for the spectacle of football when you keep giving cheap fouls away and breaking the game down and time wasting. But that's their style."
Gombau admitted it wasn't his side's best display but knew Galekovic kept it in the game.
"We got three points in a very difficult game in a place that is very difficult," he said.
"Sydney had a lot of chances but in football you have to put it it in the net and we did.
"Eugene for me is the best keeper in Australia he showed his ability not in just saving the ball but he had so many passes for us. He is very good with his feet and hands."
