Posh were almost caught napping in the opening minute when the back line was caught square as the ball was pumped forward to leave Michael Boulding in the clear.
The striker stormed into the penalty area but quick-thinking Mark Tyler advanced to spread himself and make solid save.
From that point Mansfield's Carl Muggleton was by the far busier goalkeeper although twice smart defensive work by skipper Jake Buxton averted dangerous situations created by Jamie Day from a wide role on the Posh right.
Although Muggleton did well to deny George Boyd he was powerless to stop Josh Low opening the scoring on 29 minutes.
The former Leicester man claimed his second goal of the season by forcing the ball into the roof of the net at the far post after Aaron McLean headed on.
Home penalty appeals were turned down when Martin McIntosh appeared to push McLean as he battled to get the ball away following another Muggleton save from Boyd and, in first-half stoppage time Low picked up the loose ball and blazed wide after Muggleton had kept out yet another well-struck Boyd effort.
Posh paid dearly for sloppy defending eight minutes into the second half as John McAliskey was unchallenged as he nodded in a Matthew Hamshaw cross to level the scores.
The Stags looked like getting at least a share of the points after Muggleton produced a superb one-handed save to keep out McLean's lob, but home boss Darren Ferguson rang the changes sending on Crow for the last 20 minutes.
The former Norwich youngster made the most of his chance to shine, reaching a cross from Low to restore the home side's lead, steering in a header from six yards.