The 20-year-old has signed a four-year deal at London Road and will join up with the Posh squad immediately - though he will not be able to play this season.
Goalkeeping coach Barry Richardson said: "He is one for the future but he has plenty of potential and I am looking forward to working with him."
Day broke into the Rushden team last September and had been a regular since, making 34 starts for the Blue Square Bet Premier club.
Source: PA
Source: PA