The striker scored with his right foot, left foot and then planted home a header as the lads produced a performance to remember at the Mini Estadi.
Barcelona started well but two goals in the space of four minutes - Coulthirst on 25 and 29 - turned the game in our favour.
Coulthirst's hat-trick goal on the hour gave that extra cushion and Alex Pritchard capped another impressive performance with the fourth goal on 73 minutes.
Barcelona scored a consolation goal through Cristian with two minutes remaining.
The victory moved the team level on points with Barcelona, who now top Group One on goal difference.
"We were outstanding," said coach Alex Inglethorpe.
"Barcelona bossed the first 20 minutes and I thought we were a little like a rabbit caught in the headlights, we froze, but the important thing is that we didn’t give up during that spell.
"Even though we didn’t see much of the ball and we got a bit of a pulling around in terms of them being at home and playing how Barcelona can play, we stuck at it.
"We grew in confidence and then from a counter-attack down the right, Shaq Coulthirst finished really well inside the box for 1-0. The warning signs were there because he’d got in a similar pocket five minutes before that. This time he made no mistake.
"That goal changed the momentum in the game. It rocked them and at the same time, gave us belief and confidence and four minutes later, we scored the second goal from a similar situation, the ball was loose in the penalty area and Shaq fired home this time with his left foot.
"We spoke about the ‘what-ifs’ at half-time and the key thing was that we didn’t want to limp over the line, we didn’t want to be hanging on at 2-1 and praying for the referee to blow the final whistle.
"We thought if we maintained our concentration, we’d comfortably get over the line, and that’s what happened.
"We scored the third goal after an exquisite cross from Jack Barthram found Shaq, who headed home at the far post.
"The fourth goal was a fantastic counter-attack down the right, slick play as Kenny McEvoy passed inside to Alex Pritchard and he danced around a couple of challenges and smashed home from about eight yards out.
"It was a classy goal from a player who impressed all night. The only blot on the copybook was conceding a goal two minutes from time.
"Overall, we were excellent. We were confident we could produce a performance here and the players have done it in style."
Spurs U19s: McGee, A McQueen, Barthram (Ogilvie, 74), Veljkovic, Ball, Stewart, McEvoy (Winks, 85), Bentaleb (Lesniak, 74), Coulthirst, Onomah (Dombaxe, 58), Pritchard (Michael-Percil, 85). Unused sub: Priestley.
Photos - with thanks to FCB/Alex Caparros