Havant, who ply their football in the Blue Square South division, made a mockery of the 83 places which separated them and Swansea as they opened up a 3-0 lead after just 37 minutes.
Swansea hit back through Guillem Bauza, and then home goalkeeper Kevin Scriven saved Leon Britton's weak penalty just before the break.
Gale added: "To go 3-0 up was unbelievable - you don't plan for that. We tried to keep focused but then they scored straight away and then they get a penalty.
"Kevin Scriven saved it and I think that was the turning point of the game. It was probably my hardest team-talk at half-time after that.
"It was a massive test for us to hold on for another 45 minutes and the big thing was not to concede an early goal. We did concede but then we went on to score again.
"Everyone defended for their lives and we kept our shape, eventually they ran out of ideas."
Liverpool fan Baptiste, whose 87th-minute equaliser in the first tie secured Havant a replay, said playing at Anfield on 26 January will fulfil a life-long ambition.
"It's going to be an amazing feeling going to Liverpool and seeing Steven Gerrard and Fernando Torres and playing on the Anfield pitch," he said.
And as a reward for beating Swansea and making the fourth round, Havant have been promised an end-of-season holiday to Las Vegas.