Barnsley's 2-2 draw at Huddersfield was just enough to keep Flitcroft's team in the second tier of English football thanks to a late Crystal Palace winner at Selhurst Park.
Chris O'Grady had given Barnsley an early lead, before the Terriers levelled matters on 53 through Jermaine Beckford.
Jason Scotland headed the visitors back in front before James Vaughan grabbed Huddersfield's second goal eight minutes from time.
That goal appeared to spell relegation for Barnsley but with time running out Mile Jedinak netted in south London as Palace edged out Peterborough 3-2 - thus sending Posh into the relegation zone.
And asked how he would celebrate his team's last-gasp survival, Flitcroft said: "I'll go up to my dad's grave tonight and probably spend an hour up there with dad.
"After the greatest achievements I've done in my life, I've spent them at a grave in Bolton.
"There's a lot of emotions. I just wish dad could have been here to witness that."
Flitcroft, who replaced the sacked Keith Hill in January, added: "Mission Impossible achieved.
"We've beaten some incredible football teams and Mission Impossible can only be achieved by special people.
"Forget the football, forget the quality and the disparity in this league, what we've achieved has been done by a group of special people and a special management team I got in from day one.
"I didn't trust just anyone to step up with me so I got Micky Mellon and Martin Scott and they've driven me and we've driven this team. What we've achieved is quite remarkable."