The Tykes secured a place in the fifth round with a 1-0 victory at Hull thanks to a 70th minute goal from Chris Dagnall.
Flitcroft, who has presided over three wins in five games since taking over as boss, said: "It's a great feeling to win in the FA Cup, and we have that feel-good factor and momentum.
"Our form is down to a group that wants to be part of something special. In spite of where we are in the league, we have an opportunity to put things right.
"I love the FA Cup and everything about it. It's a special competition.
"Some of the players in the dressing room have great FA Cup memories, and the fans will go back down the M62 with their own memories. I want to go as far as we can in the FA Cup as I'm a massive believer in it.
"We were really poor in the first half. Hull dominated and deserved to go in at the break two or three goals up.
"But I said to the players at half time that in the second half, when we do get a good spell, we need to score.
"Out of possession we were very good. We were resilient and determined. In possession we were not very good and there was a lack of quality, and we do need to improve."