Two goals in four second half minutes from Safuwan Baharudin and Josh Kennedy cancelled out Pablo Sanchez's first half opener and substitute Iain Ramsay added a late third as City took the spoils to take over sixth spot from Brisbane Roar. Adelaide, meanwhile, stays third.
The first chance to fell to City when the ever-dangerous Aaron Mooy found Jack Clisby in the box and his powerful strike brought a reaction save from Reds keeper Eugene Galekovic.
The City left-back was in the thick of it again three minutes later, flashing another drive just wide after being picked out by Mooy again.
Having been pinned back for long periods, the Reds broke out to take a 32nd minute lead when Sergio Cirio took a pass from Michael Marrone and flicked the ball into the path of fellow Spaniard Pablo Sanchez who, with time and space, guided a left foot shot past goalkeeper Tando Velaphi.
Five minutes later Cirio skipped across the 18-yard box before firing off a shot which brought a smart stop from Velaphi.
With substitute David Williams pulling the strings, City hit back six minutes into the second half to break a 339-minute goal draught when Mooy's corner deflected into the path of Singaporean signing Baharudin, who swept the loose ball high into the net for his first A-League strike.
And it got worse for the Reds in the 55th minute when Robbie Koren crossed for former Socceroos striker Kennedy to send a thumping header past Galekovic to give City the lead.
Galekovic then denied Kennedy his second with an instinctive stop as the Reds found themselves on the ropes.
Massimo Murdocca came close to adding a third in the 68th minute, but his shot curled wide.
But he made amends in the 87th minute, taking a pass from Williams to square the ball back across the face of goal for substitute Ramsay to stab home the third on his birthday.
City coach John van't Schip admitted he felt the confidence drain from his players after they went down a goal, but he was delighted with the way they fought back after he shuffled his formation during the break.
"After that, I think we deserved to win.
"We were dominating ... there were a few good moments in scoring and maybe we could have had an even greater scoreline but, of course, we're happy to be 1-0 behind and ending up 3-1," Van't Schip said.
Adelaide missed the chance to go top of the ladder with the loss and Gombau lamented his side's inability to cope with the changes he made to his structure due to injury and suspension.
"When you don't have all the squad, you need to put players in who aren't playing those positions (normally) - it's difficult.
"This affected the team a little bit but, in the end, it's what we had and we prepared as best as we could - it's no excuse," Gombau said.
