In his first start for the Reds, Stefan Mauk whipped in a cross deep into the Sydney penalty area, only for Seb Ryall to clear the ball straight into the legs of Cirio, sending it past keeper Vedran Janjetovic.
Isaias almost made it 2-0 just after the quarter hour but his 25-metre free-kick smashed onto the cross bar with Janjetovic beaten.
Sydney responded and dominated for the rest of the first half and were rewarded with an equaliser on 37 minutes.
Filip Holosko delivered an inch-perfect cross for Matt Simon whose glancing header nestled into the bottom corner of the net.
The visitors remained in control before Cirio helped United retake the lead with his second on the stroke of halftime.
After a mazy run from Bruce Kamau, the little Spaniard took a touch and shoot from just inside the area with Janjetovic only able to get his finger tips to the low drive.
But, there was to be more late drama before the break as Holosko made the most of some sloppy United defending to equalise three minutes into first-half stoppage time.
Milos Ninkovic dinked a delightful ball over the Reds' defence and Elsey's attempted clearance only assisted Holosko who made no mistake from close range after the ball sat up nicely.
Sydney did most of the attacking in the second half with Holosko and Simon looking the most dangerous, although the pair was largely restricted to half chances.
Simon nearly gave the Sky Blues the lead on 75 minutes with another powerful header after a teasing corner but his effort was just wide.
Minutes later, Milos Dimitrijevic had a rasping 30-metre shot tipped over the bar by Reds keeper Eugene Galekovic.
Adelaide coach Guillermo Amor said he was satisfied with a point from the tough battle against the Sky Blues.
"I prefer to win.
"We prefer to win, but after the game and the analysis, it's not (a) bad (result)," Amor said.
Sydney FC coach Graham Arnold applauded his side's response after twice falling behind.
"It was a good game of football.
"I thought we showed a lot of character tonight.
"I'm very happy with the way we played," Arnold said.
- AAP