Roar had gone five matches without a win at the Suncorp Stadium before Solorzano's late strike secured a 3-2 victory for the home side.
After a quiet opening, Wellington registered the first significant attempt of the match through Roy Krishna, but his shot was never going to trouble goalkeeper Michael Theo, who was making his first A-League appearance of the season.
A lack of goalmouth action followed that flurry of chances, but shortly before the break, Jerome Polenz, who was making his debut for Brisbane, narrowly missed the target after combining with Borrello.
Brisbane began the second half on the front foot, with Solorzano having a header well saved by Glen Moss, but five minutes after the restart, Wellington went ahead when Krishna converted from close range.
Burns almost doubled Wellington's lead three minutes later only to be denied at the near post by Theo, and that intervention proved important as James Donachie found the equaliser after connecting with a corner.
The leveller boosted the home side, and after Luke Brattan had gone close, his side edged in front with just 14 minutes remaining when Borrello latched on to a through-ball before finishing past Moss in the Wellington goal.
However, as the game entered the final 10 minutes, Wellington restored parity after Kenny Cunningham charged down a clearance from Theo before tapping home into an empty net.
It appeared as though the match would peter out, but deep into added-on time, Solorzano got on the end of a Polenz cross to break his team's winless streak in Brisbane.