It was Barnsley's third draw in a row as they stretched their unbeaten run to four matches.
The result ended Doncaster's run of three defeats - but th ey nearly snatched a win in the last minute when Richie Wellens curled his shot from the edge of the area, just wide of the far post.
Flitcroft said: "I really wanted to win this game and I feel frustrated like a fan that we haven't taken all three points today.
"We had the quality to finish them off. Today we missed an opportunity against a side who came with a defensive plan."
He added: "We definitely wanted three points but we didn't work their keeper enough.
"In the first half we should have come in three up and we had more chances in the second half."
Former Barnsley goalkeeper Ross Turnbull saved an 18th-minute penalty from Chris O'Grady.
Flitcroft said: "It was something you don't see often, O'Grady missing a penalty.
"I've said to him that he has now joined the list of good players who have missed one."
Barnsley thought they had won their second penalty of the match when Paddy McCourt went down in the area after a run - but referee Kevin Friend gave a free-kick to the visitors.
Flitcroft added: "It was a penalty; I think that if he hadn't given us a penalty in the first half then he would have given us that. Paddy was about to shoot and he wouldn't have gone down."
Doncaster boss Paul Dickov was pleased with a point.
He said: "We put on a good defensive display with an inexperienced back-four.
"I can't argue about the penalty in the first half, but what I can argue about is that we should have been given one when Chris Brown went down.
"The thing that disappoints me is that we didn't work the ball around when we won it.
"Richie (Wellens) should have scored. But we have come here with a depleted squad and got a point so that pleases me."