Flitcroft was appointed caretaker boss following Keith Hill's departure and after defeat at fellow strugglers Peterborough in his first game in charge, the Tykes have clinched back-to-back victories.
A late goal saw off Burnley in the third round of the FA Cup and they then clinched their first home league success since September 1 with an uplifting 2-0 derby win over Leeds on Saturday.
A club statement said: "It is clear that David has the passion, personal dedication and the commitment of the players that are essential if we are to maintain our Championship status.
"To be successful, David will also need the support of the fans and we ask that all those who care about Barnsley FC show unity and get behind David and the Reds."
Flitcroft, who had been Hill's assistant at Oakwell and also at Rochdale, revealed after the win against Leeds that he would not turn down the job should it be offered.
"I have found out a lot about myself in the last 10 days. It could change the way I look at myself for the rest of my career," he said.
"When you lead a group and they are all playing for you, it's different. I'm not going to walk out on that group of players. If I get sacked, I get sacked, but I've been taught well and you don't walk out on people."
Barnsley have been turned down by two managers since sacking Hill after the home defeat to Blackburn on December 29.
Sean O'Driscoll, axed by Nottingham Forest just hours after their Boxing Day victory over Leeds, was first to reject the Tykes' advances and then Inverness boss Terry Butcher declined the chance of a return to English football and opted instead to stay north of the border.