The 12-1 shot, winner of last year's Irish St Leger on the flat, beat Cheltenham Champion Hurdle runner-up My Tent Or Yours by a length and a half.
Another Mullins-trained horse Arctic Fire was third as he took the trainers' title lead over Gordon Elliott.
Patrick and Willie celebrated another Grade One win with Bacardys.
Bacardys won the Champion Novice Hurdle, and the pair followed up with Montalbano in the novice hurdle.
Trainer Mullins sent five runners to post for the Champion Hurdle contest, although that number was reduced after an unsettled Diakali was withdrawn at the start.
Of the remaining trio, Wicklow Brave was the outsider but took over the lead from Identity Thief in the back straight and never looked back.
Jockey Mullins punched the air as his mount, wearing blinkers for the first time, clinched victory.
My Tent Or Yours took another big-race runner-up spot after being pipped by Buveur D'air both in the Champion Hurdle at Cheltenham and at Aintree earlier this month.
Arctic Fire's third-placed finish provided Mullins with a further boost in his battle with Elliott for the Irish trainer's title going into the final day of the season.
The Gordon Elliott-trained Supreme Novices' Hurdle winner Labaik was fourth with Mullins first string, Vroum Vroum Mag, well beaten.
"He was nearly stood at the start, but when he was going forward and jumping fantastic I didn't want to bring him back," said winning jockey Mullins.
"I didn't think we were going a mad gallop and I said I'd take a chance and just let him enjoy himself."
Wicklow Brave's St Leger victory at the Curragh last September gave Irish champion trainer Mullins his first Classic success.
BBC horse racing correspondent Cornelius Lysaght:
What a potential fairytale end to the Irish season if Willie Mullins can hold on to this lead gained thanks not to Walsh or Townend, but his amateur-jockey son Patrick.
That said, Mullins junior's experience and skills in the saddle make him a professional in all but name, and the wily ride on Wicklow Brave, knowing his stamina and taking command at halfway, was a joy.
It's not over yet: the gap could easily be bridged back the other way on Saturday, but I must apologise for doubting Mullins and team in midweek. They've been magnificent.
Wicklow Brave won the Punchestown Champion Hurdle as jockey Patrick Mullins rode a treble for his title-chasing trainer father Willie.