Star striker Jeronimo Neumann slotted a goal for the Reds in the 58th minute - his fifth this season - to deny first-year Heart coach John Aloisi in his home state.
The South Australians' day then got better when closest rivals Central Coast Mariners drew 2-2 at Melbourne Victory.
The Mariners twice went ahead through Daniel McBreen (4, 22) but were pegged back on each occasion by Andrew Nabbout (10) and Mark Milligan (40).
Central Coast had Dutch defender Patrick Zwaanswijk sent off on 50 minutes but they held on before Marcos Flores saw red for the hosts in injury time.
That ensured a good day for Adelaide, who bossed the majority of the first half but were unable to capitalise early as the visitors held their nerve under pressure.
Cassio pushed a good-looking shot over the crossbar in the 22nd minute to set the home side into motion.
From close range a minute later, Argentine duo Marcelo Carrusca and Neumann were denied by some desperate scrambling from veteran Melbourne goalkeeper Clint Bolton.
Thompson gave Melbourne their best first-half chance, breaking to the left in the 35th minute, but his miscue was comfortably mopped up by Adelaide custodian Eugene Galekovic.
Three minutes later, the Reds missed a golden opportunity to break the deadlock.
After a Simon Colosimo defensive error, Jeronimo, who had been relatively quiet, had a shot which Bolton just got his outstretched right hand to.
In the 45th minute, Jeronimo pulled the trigger again, driving a fierce ball just wide before Evan Kostopoulos, in time added on, unleashed a shot from the flank which whizzed past Bolton's right post.
The signs were better for the visitors early in the second half with Garcia twice cutting inside the box and getting behind Adelaide's defence, but despite his efforts the deadlock remained.
United weathered the brief Heart storm and resumed control with Jeronimo working his way into the contest.
The home side were finally rewarded for their all-round ascendancy when they took the lead in the 58th minute courtesy of an uncharacteristic defensive blunder from seasoned central defender Colosimo.
The former Socceroo coughed up an errant kick which was won by Vidosic. The ball then fell to the feet of hungry Jeronimo, whose shot from outside the box took a deflection off Patrick Gerhardt and left Bolton flat-footed.