Pressley lavished praise on his team as they reversed the form book with a first away win since January.
A first-half goal from Harry McKirdy set the visitors on their way before they took complete control in the second half, adding to the scoreline through Jack Bridge's penalty and Nathan Thomas' fine solo goal.
He said: "It's an amazing performance, it shouldn't be underestimated. They made some changes but (Barnsley) had a really strong team out tonight, a Championship team.
"Although they had a couple of moments in the game, I think the score wasn't a true reflection. I think we could have won by more, over the course of the 90 minutes.
"It was a truly amazing effort by the players. I spoke to them after the game on Saturday and we did a lot of work between then and now. We have to pride ourselves on nobody working harder than us.
"When you go through the pre-season we go through, you do it for a reason. The reaction of the players was amazing tonight. To a man, there wasn't a disappointment."
Barnsley boss Daniel Stendel said: "Nobody expected this performance from us today. In the end, Carlisle won 3-0 and it was deserved. It was hard for us today.
"I think the quality from (individual) players was good enough to win this game but the quality (as a) team was not good enough.
"We missed a lot of things and we have three days to play Charlton (and must be) so much better than today. Only then do we have a chance to win this game.
"They need time to improve. We made too many mistakes in the offensive (areas), we lost a lot of easy balls and this is (what happens) when you want it too much."