Jones becomes the Red Devils' first signing since securing promotion from League Two. The 25-year-old is set to replace Scott Shearer as number one following the latter's move to rejoin former boss Steve Evans at Rotherham.
A virtual ever-present for Posh this term, former Exeter man Jones made 37 appearances in all competitions.
Crawley chief executive Alan Williams said: "Paul is an outstanding acquisition for the club. He is a commanding presence and has lots of experience at Championship level.
"We are delighted he is here and we think our supporters will enjoy watching him and we look forward to him enjoying a long and successful career with the club."
Jones added: "I have followed the club's success over the last two years and it didn't take me long to decide I wanted to be part of an ambitious club like Crawley. It's a great set-up with Steve Coppell and Sean O'Driscoll involved and I can't wait to get started in pre-season in July."