The Reds seemed to be cruising into the interval on Friday night with a 2-0 lead before the visitors pulled one back in first half stoppage time.
Western Sydney scored twice more in the second half to claim maximum points and end United's undefeated start to the season.
On six minutes, Dylan Pierias played a one-two with Archie Goodwin before his cross almost caught out Lawrence Thomas, with the Wanderers keeper scrambling to keep the ball out.
Ten minutes later, Marcus Antonsson let fly from the edge of the area, forcing an impressive diving save from James Delianov.
Adelaide eventually wrestled control of the contest and broke the deadlock on 31 minutes thanks to Ethan Alagich who chalked up his maiden A-League goal.
Stefan Mauk released Ben Folami wide on the left, and Folami slipped in Alagich behind the Wanderers' defence.
The Adelaide midfielder blasted first time from a tight angle, beating Thomas at the near post and rifling his shot into the roof of the net.
United doubled their lead just three minutes later with Goodwin's fifth goal of the season.
Zach Clough took possession 30 metres from goal and whipped in a cross from the right. The delivery was inch-perfect, finding Goodwin on the penalty spot, who nodded it past Thomas.
Western Sydney pushed to claw a goal back and on 41 minutes Jack Clisby stung the palms of Delianov, driving a low cross through the six-yard box.
With the interval just seconds away, the Wanderers managed to find the net.
Nicloas Milanovic crossed from the right to Antonsson at the back post. The Swedish striker powered a header past Delianov who managed to get a hand to the ball but couldn't keep it out.
Four minutes after the restart, the Wanderers were level.
Clisby whipped in a corner which the Reds failed to clear and Brandon Borello was on hand to pounce, slamming the ball into the back of the net.
The Wanderers' comeback was complete on 82 minutes when substitute Kraev drilled home the winner from the edge of the area.
Western Sydney robbed Adelaide of possession on half way and broke forward with Dylan Scicluna supplying the decisive pass.
Kraev surged forward with the United defence sitting off the attacker before he fired low and hard under Delianov.
There was more late drama, with Wanderers' coach Alen Stajcic sent from the bench by referee Daniel Cook in the dying minutes.
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