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The Reds dominated the reigning champions for long periods, with goals from Bruce Djite and Sergio Cirio either side of the break handing them a well-deserved 2-0 win at a wet Coopers Stadium.
“The boys were great, we kept possession and held onto it for long periods of time and we made them work,” Galekovic said.
“It was a great performance from all the boys. Defensively we did very well. I think they had one shot on target and we had numerous chances.
“We deserved the victory, we deserved it and we move onto the next round.”
Waiting for Adelaide in the last eight is Sydney FC, who saw off Sydney United 3-1.
It’s the third straight round the Reds will meet A-League opposition in the Cup, after wins against Wellington Phoenix and now Brisbane.
“It’s another test. These games are do-or-die so we know if we lose we’re out,” Galekovic said of the upcoming trip to Sydney FC.
“We’ll give it everything, we’ll prepare well and we look forward to it.
“Everyone wants to win the first FFA Cup and we give ourselves a good chance. We’re in the quarters now and its good preparation for the A-League season as well.”