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Calum Kavanagh's sixth goal of the season gave Latics the lead in the 66th minute, only for Davis Keillor-Dunn to score against his old club and draw the League One side level.
But, despite manager Micky Mellon being sent to the stand after an altercation with Callum Slattery, Latics retained their focus and, after taking the game to extra time, won it with a Will Sutton header.
"It was a real end to end game and I think it would have been a fantastic game to watch, and I thought it was a game that we deserved to win," said Brabin in the post-match press conference, with Mellon's red card ruling him out of media duties.
"I think that was a real big show of character. They're a good side, they've got good players, they've got a new manager come in that they're wanting to impress. I thought we deservedly took the lead and they've equalised and we want to see a little bit of grit, a bit of character, and I thought we showed that.
"The fans were behind us today, it was a great atmosphere and I think they could see as well the fight on the pitch and determination. It was a great cup tie."
And Brabin praised match-winner Sutton for his roll in it.
"He was brilliant today, a real rock," said Mellon's number two.
"I'm probably Will's biggest critic at times but that's because I've got a lot of belief in him that I think he's going to be a really good player. He's got all the attributes to be a really good defender and he showed that tonight."