All five goals came in a dramatic second half at Suncorp Stadium, with Roy Krishna breaking the deadlock before James Donachie and Brandon Borrello responded for the hosts.
Kenny Cunningham looked to have earned Wellington a share of the spoils, only for Solozarno to win it in stoppage time.
Phoenix would have gone second with a victory, but remain fourth, while Brisbane are now four points off a final series spot.
Krishna opened the scoring in the 50th minute with a toe-poked effort for his sixth of the season, but Donachie's header levelled matters on the hour mark.
Borrello's cool finish past Glen Moss made it 2-1 with 14 minutes remaining, but Cunningham fortuitously equalised five minutes later as he was hit by Michael Theo's attempted clearance.
It was Solozarno who stole the headlines, though, as he turned in Jerome Polenz's cross to settle matters at the death.
