Costa Rican duo Carlos Hernandez and Kenny Cunningham netted before half-time to set up the crucial win at Eden Park.
Previously unbeaten in seven matches, a frustrated Adelaide outfit will return to Australia having missed the chance to move to fourth, with Awer Mabil's last-minute consolation coming too late for the tired visitors.
Busy all afternoon, Hernandez scored the first goal from a penalty and created the other in the shadow of half-time for Cunningham, who produced a slick finish.
The result boosts the Phoenix to just a point behind Adelaide, who had started the round in sixth place.
Elsewhere, a last-minute strike from Adam Taggart secured the Newcastle Jets a thrilling 2-2 draw with the Western Sydney Wanderers in an action-packed clash at Hunter Stadium.
Newcastle took the lead thanks to a wonder strike from Taggart midway through the first-half before the Wanderers levelled with a scintillating set-piece from Aaron Mooy on the stroke of half-time.
Western Sydney skipper Michael Beauchamp put his side into the lead just after the hour mark when he was in the right place at the right time to deflect a Josh Brillante clearance into the Jets' net before Taggart snatched a share of the spoils when he poked home in the 90th minute.
The draw sees the Wanderers maintain second spot in the A-League while the Jets are six games without a win.
