The visitors soaked up long spells of Hull pressure before snatching the lead with 20 minutes remaining to give Flitcroft a third win in five games since taking over as Barnsley manager.
Flitcroft 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."
He continued: "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."
Hull manager Steve Bruce admitted his team has "bigger fish to fry" as they attempt to secure promotion to the top flight, but he could not hide his disappointment with his side's lack of cutting edge in attack.
"We have 18 games left in the Championship now without any distractions, but it's always disappointing to go out of the cup," Bruce said. "If you get a favourable draw before you know it you can be in with a chance."
Bruce, who felt his side should have been awarded a penalty in the first half when Kelvin Etuhu appeared to bring down Stephen Quinn, played down his chances of bringing in a new striker to bolster his side's attacking ranks before the transfer window closes.
"If I'm honest, I'm not confident with the way it is looking at the moment," Bruce said.
"We have let a couple of the younger players go out on loan to get some experience, but we are not close to anything at the moment. We'll have to see if we can get anyone else in, but they have to be the right player."