A beautifully created early strike from transfer-listed frontman Aaron McLean proved to be enough to see off a Rangers side that finished the match with nine men after the second-half dismissals of skipper Mikele Leigertwood and Peter Ramage.
Former Stockport County and Motherwell chief Gannon is Posh's third boss of the season and this result meant he equals the tally of wins managed by predecessor Mark Cooper in a two-and-a-half month spell at the London Road helm.
McLean was left with the simple task of slotting home from six yards after skipper George Boyd had danced round two tackles inside the area.
It was McLean's eighth goal of the season but the first since he slapped in a transfer request in December.
McLean and strike-partner Craig Mackail-Smith posed constant problems to a shaky Rangers rearguard as the arrival of Gannon breathed hope into Posh's battle against the drop.
Goalkeeper Carl Ikeme had to dash from his line to beat Mackail-Smith to a classy Boyd pass before needing two attempts to hang on to a 20-yard blast from wing-back Chris Whelpdale.
Recalled Tommy Rowe - a man who graduated in the game at Stockport under Gannon's supervision - should have done better with a good headed opportunity while Leigertwood provided home keeper Joe Lewis' sole test of the opening period.
He was forced to make a tougher save after the interval when Matthew Connolly's drive bounced through a crowd of bodies, but Posh were still well on top.
Their grip on proceedings was strengthened when Rangers were reduced to ten men. The referee had no hesitation in pulling out a straight red card after Leigertwood had launched into a two-footed challenge on Rowe – and he wasn't alone in taking an early bath.
Ramage also received his marching orders when picking up two cautions in the space of nine minutes, both for fouls on Rowe.
Despite their numerical disadvantage Rangers piled on the pressure in the dying moments, but Posh stood firm to keep a first clean sheet for almost two months.