The 22-year-old has made a big impact since his arrival from non-league Rushden & Diamonds in August, scoring nine times in 33 appearances.
Posh manager Darren Ferguson told the club's official website: "We're all delighted. He's done ever so well since he came. He's settled well and I believe he has real potential.
"In his first season at this level he has proved he can handle it and I believe he can play in the Championship no problem at all.
"It's a lengthy contract. Lee knows our ambitions and I think it's important we get it out of the way because coming into March we need everybody focused on the job in hand."
Source: PA
Source: PA