Victory came at a cost, though, as US talisman Christian Pulisic came off injured before the half-hour mark after close attention from defender Jason Geria.
A first heavy challenge by Geria had Pulisic clutching his right ankle, and while he got back on his feet, he was soon brought down again when Geria clipped his heel from behind to earn a yellow card.
After being examined by trainers, Pulisic trudged off and was replaced by Diego Luna.
Coach Mauricio Pochettino said the AC Milan attacker, who was wearing the captain's armband in Tim Ream's absence, had felt "something in his hamstring".
"Tomorrow he will fly to Italy. Tonight we will assess (him) but we cannot say anything at the moment," said the Argentine.
Pulisic's blow compounded a poor start for the hosts who were a goal down at Dick's Goods Sporting Park after 19 minutes.
Australia full-back Jordan Bos capitalised on static defending after his own throw-in to weave through traffic in the box and thump in a volley from close range.
But Wright levelled the match in the 33rd minute with a clever toe-poke at the near-post, latching onto a through-ball from Cristian Roldan.
The Coventry City forward then curled in the winner five minutes after the restart, turning Cameron Burgess in the box after the Australians were caught out by Roldan's long free kick down the right channel.
Wright's brace ended the Socceroos' seven-match winning streak while ensuring the US finished the October international window on a high after being held 1-1 by Ecuador in Texas last week.
"Our performance was good but I think the team is ready to perform much better," said Pochettino.
"Am I happy? Yes. But I wanted to see more, to see individual players do better. That is my competitive blood.
"We cannot concede the goal we conceded at a World Cup. If we start a game like that, it will be difficult to get another chance."
Australia, meanwhile, suffered their first defeat in more than a year, having been unbeaten in 12 matches and 11 under Popovic.
Both sides rejigged their lineups from their previous friendlies.
Pochettino started with Wright and Pulisic up front, while Popovic restored goalkeeper and captain Mat Ryan to the starting 11 after using Paul Izzo in the 1-0 win over Canada.
The Socceroos' youthful attacking duo Nestory Irankunda and Mohamed Toure came off the bench after the hour-mark but were unable to conjure the equaliser to sustain Popovic's unbeaten record.