Chris Coleman's ambitious Sky Blues pulled off a fourth victory in five outings to keep alive hopes of getting into the play-offs.
But they first had to surviving a bright Posh opening when Chris Whelpdale fired narrowly wide after just 10 seconds.
Then Gaby Zakuani, back for bottom of the table Posh after a lengthy injury absence, headed over from a swirling Lee Frecklington free-kick.
Coventry grabbed a 24th minute lead very much against the run of play with Martin Cranie's 40-yard punt forward releasing Jon Stead, with the 6ft 4in tall striker muscling his way past Craig Morgan to fire past goalkeeper Joe Lewis from the edge of the area.
Former Norwich man Sammy Clingan was booked for a 32nd minute foul on Whelpdale, but the Northern Ireland international rescued City four minutes later after heading a goal-bound Charlie Lee effort against the underside of the crossbar following a corner from Boyd's replacement Dominic Green.
A tame second half saw Coventry trying in vain for a further goal to put the result beyond all doubt.
Zakuani denied the lanky Stead and James McPake was woefully off target with a header from Michael McIndoe's cross.
Clinton Morrison could hardly believe his luck when finding himself in possession 10 yards out as Posh waited for an offside decision that never came.
Lewis came to the rescue with a point-blank save to keep the home side in the hunt.
Despite seeing plenty of the ball, Posh created little and even manager Jim Gannon's 74th minute introduction of out-of-favour frontman Craig Mackail-Smith failed to come anywhere close to sparking a late equaliser.