Cornishman Day, who has worked his way through the London Road youth ranks, delivered a stunning final blow to sorry Bury with a classy individual strike.
The 20 year-old collected a 64th minute short corner, tricked his way past two defenders and guided a crisp shot past keeper Neil Edwards.
Posh needed only six minutes to seize the initiative when Northern Ireland front man James Quinn pounced on a rebound to break the deadlock.
Strike partner Calum Willock left the Bury defence trailing in his wake before seeing his shot saved by veteran keeper Edwards, but Quinn was on hand to rifle the follow-up into the roof of the empty net.
It was Quinn's fourth goal of an indifferent personal season in his first starting appearance since returning from an unsuccessful loan spell at League One club Bristol City at the beginning of December.
And the home side received a helping hand from Bury defender Colin Woodthorpe when doubling their lead five minutes before the interval.
Day dashed down the left flank and delivered a teasing cross that Woodthorpe headed past the stranded Edwards and into his own net.
And the visitors were killed off within 90 seconds of the second period starting when Sagi Burton further increased the Posh lead with a header which found its way past Edwards with the aid of a deflection off Andrew Parrish.
Veteran centre-back Dave Challinor slotted a Bury consolation 18 minutes from time following a corner Posh felt should not have been awarded before home keeper Mark Tyler made a smart late save to deny substitute Jake Speight in a one-on-one situation.