Without a win since their opening-round victory over Perth Glory and with pressure quickly mounting on coach Josep Gombau, who lashed out at local media earlier this week for the second time this season, the Reds produced a superb display in accounting for the reigning champions.
Sergio Cirio laid on goals for Bruce Djite and Fabio Ferreira in the first half to put Adelaide in control.
Cirio then scored his side's third goal after 54 minutes and substitute Jeronimo Neumann added a fourth at the death to round off a thumping win.
Wellington's Carlos Hernandez converted a disputed penalty two minutes after half-time to cancel out Matt Smith's first-half opener.
In Saturday's other game, Mark Bridge came back to haunt his former club Newcastle Jets, his header giving Western Sydney Wanderers a 1-0 win at Hunter Stadium.
Bridge, who etched himself into Newcastle folklore by scoring the winning goal in the Jets' championship winning 2007-08 season, scored in the 33rd minute of a gripping encounter.
