The 22-year-old worked under Grant McCann at Posh.
His arrival boosts competition for places in midfield following Luke McCullough’s loan switch to Tranmere Rovers.
“I’m delighted to be here, the manager gave me a call and expressed an interest and it was a move that appealed to me,” Anderson told Rovers’ official website.
“I like to attack and win the ball back, for me it’s about showing what I can do and getting around the field as much as possible.
“I’m raring to go and I’ve already trained with the lads, they’ve made a good start to the season so hopefully we can keep that going.”
Like McCullough, Anderson - who is Peterborough’s longest serving player - has had his injury problems in recent years and has been restricted to 50 appearances in the last three seasons.
Despite only featuring four times for Posh in 2018, McCann says his latest recruit is ready to play.
“Jermaine is a good player, the fans will like him and he has really good energy in the midfield area,” said McCann.
“He was someone we looked at in the early part of the window and wanted to get the timing right on it, I think it’s the right move for us.
“He’s ready to come into the squad whenever he’s called upon, his physicality is something different to what we have in the squad.”
He added: “He can keep the ball and move people around, it wasn’t long ago that he was being linked with a big money move.
“He can complement what is already here, he’s very forward thinking but can play in a number of different systems.
“He knows a few of the lads already from different places so it will be easy for him to settle in, we’re delighted to be able to bring him here.”