Flitcroft, who was sacked by Barnsley two weeks ago, saw a Dean Howell own goal cancel out Daniel Nardiello's penalty.
Northampton winger Chris Hackett also missed a penalty to deny Aidy Boothroyd's side a crucial win.
A sweeping counter-attack almost brought early reward for Bury but Shaun Harrad fired wide after Matt Duke spilled Chis Sedgwick's cross.
But the Shakers went in front in the 23rd minute when Ricky Ravenhill fouled Craig Jones inside the area and Nardiello thumped the spot-kick home.
Despite the hosts' dominance, Northampton scored a shock equaliser in the 29th minute when Howell headed Hackett's free-kick into his own net.
Then Hackett missed from the spot in the 45th minute, blazing over after William Edjenguele had upended Izale McLeod.
Duke made a stunning reaction save just after half-time, keeping out Nardiello's first-time volley from a Tommy Miller pass.
At the opposite end, Brian Jensen superbly denied McLeod at point-blank range and Mathias Doumbe missed a golden chance late on.