Cooper, 40, left his job as Kettering manager following Saturday's 2-0 Conference victory over Cambridge and has signed a three-and-a-half year contract at London Road.
Posh parted company with former boss Darren Ferguson by mutual consent on Tuesday with the club at the bottom of the Championship table.
"I'm sad to leave Kettering and it would have been nice to finish what we started, but this is a fantastic opportunity," said Cooper.
"I'm just going to go in and try and lift them a bit and try and get some results as quickly as possible.
"It's a big task but one I'm looking forward to. I've been used to winning games for the past two and bit years and I want that to continue."