Trailing 2-0 inside 25 minutes, the forward produced two fine finishes either side of half-time before Jack Munns completed the job just after the hour.
"Bristol dropped deep defensively and that gave us a problem to solve tactically," said coach Alex Inglethorpe.
"It also meant we were susceptible to the counter-attack and we were caught out twice in the first half. The players will learn in time how to defend that situation better.
"I was delighted though that we kept our identity, we didn't panic and believed that the way we play is the right way. We didn't surrender that.
"We got back into it in the 36th minute when Jack Munns played a reverse pass to send Darren McQueen clear and he finished well one-on-one.
"The equaliser came 10 minutes into the second half when Nabil Bentaleb pressed a defender and managed to get a block on an attempted back-pass, Darren got there and finished well again.
"We scored the winner after 65 minutes when a good move down the left ended with Grant Ward crossiing and Jack timing his run to steer the ball home.
"At 3-2, Bristol came out a bit more and looked more dangerous, but we held on.
"It was a strange game in many ways as we were afforded so much possession.
"The players will come up against different styles of play throughout their careers and it's all about solving these tactical problems."
Spurs U18s: McGee, Stewart, A McQueen (Zapata-Caicedo, 65), Ball, Barthram (Modeste, 35), Bentaleb, Lameiras, Munns, Ward, Coulthirst, D McQueen (Oduwa, 60).