The visitors missed numerous opportunites at Kenilworth Road before a late goal for the home side saw the spoils shared.
"We controlled and dominated the home side really well. I thought our tempo was good, our quality was as good as I've seen it, but we've not let it slip in the last five minutes, we've let it slip in the first half," Flitcroft told Sky Sports News.
"To completely dominate a team like Luton and not score the goals... the first half we got a lot of good interchanges and good quality, but we didn't hit the target enough, didn't have the last bit of quality."
Bury continue their League Two campaign this Saturday at home to Plymouth Argyle.