The home side were on the backfoot for the majority of the game, but a dramatic seven minutes, which included three goals and a sending-off, resulted in Brisbane progressing.
The visitors started brightly by throwing waves of attack at Brisbane's defence, one of which ended up at the feet of Kosta Barbarouses, who managed to get a shot away, but Jamie Young produced a decent stop.
The defender's attempt rebounded out to Carl Valeri, who struck towards goal, but he missed the target and the ball trickled wide of the post.
Victory continued to pile on the pressure as Barbarouses got on the end of a failed headed clearance by Thomas Broich before delivering in a low cross, but Young came off his line to collect the ball.
Oliver Bozanic tried his luck from a set piece as he aimed for goal from 20 yards out, but his attempt went straight into the hands of Young, who was by far the busier keeper in the first half.
Brisbane's attacks were few and far between in the first 45 minutes, but they came close through a counter-attack when Corey Brown sent a nicely-weighted through-ball to Broich, whose cut-back was cleared.
It took 69 minutes for the home side to get two shots on target, but both of them failed to trouble goalkeeper Thomas Lawrence, who comfortably handled the attempts.
With the clock ticking down, Brisbane tried to create more attacking threat and they came close to breaking the deadlock with a set piece.
Corona delivered the ball into the box and he managed to find Broich, who leaped in the air to reach the ball, but his header drifted over the crossbar.
The forward found the back of the net with a 12-yard volley from a corner kick, but Brisbane responded instantly as they found an equaliser one minute later.
The goal came from Matt McKay, who met the ball as it drifted towards the far post, and he flicked it in with a slight touch of his boot to spark the game into life.
In the fourth minute of injury time, Broich completed the dramatic comeback by heading the ball into the back of the net off a corner kick.
Brisbane will face Adelaide United in the semi-finals.