The 20-year-old forward, who scored 16 goals while on loan at Southend last campaign, has signed a four-year deal with the London Road outfit.
Although the fee has not been disclosed, Posh chairman Darragh MacAnthony said on Twitter: "Delighted to finally land our No 1 summer target and smash our transfer record with young Britt Assombalonga signing 4 year deal today!"
The club's previous record transfer fee was £1.1million, which was paid to Crawley for Tyrone Barnett in 2012.
Peterborough had been on the hunt for a new striker following Dwight Gayle's move to Premier League new boys Crystal Palace, and manager Darren Ferguson was thrilled to get his man.
He told the club's official website: "Everyone is aware we have been looking for a striker and this is the one that we have been after.
"Since Dwight left, we have been after him. We identified him, we have had him watched, he did very well at Southend and Watford thought very highly of him.
"I had a good chat with Gianfranco about him. We thought initially it was going to be a loan, but the chairman has come up with the money and that is fantastic for me. He has once again backed me. It was a position we needed to fill.
"He has a lot of potential the boy, he is strong physically, good pace, good finisher and a really nice lad. He ticks all of the boxes. He has improvement in him, but I don't think people will be too surprised, because it fits our criteria."