The 21-year-old joined Sunderland's academy as a six-year-old and made seven first-team appearances in all competitions, though he was not part of the club's Premier League squad last season.
Peterborough boss Luke Williams said he was delighted to win the race for the midfielder.
"Harrison is an ambitious and talented young player who has had a great start to his career at Sunderland, with first team experience and being in and around the first team for a couple of years," Williams said.
"He had a lot of different options for the next stage of his career so we are delighted that he has chosen us and we are looking forward to working with him."
Jones said the move happened quickly once interest emerged.
"It has all happened quite quickly but when I heard of the interest it was definitely something that excited me," he said. "There are a lot of big clubs in League One but we are a big club too and I can't wait to get started."
Jones is Peterborough's second signing of the summer, following the arrival of defender Evan Weir after his release by Walsall.
The deal represents a step up for Jones, who moves from Premier League academy football into the professional game in England's third tier, where Peterborough have established themselves as one of League One's bigger clubs.
Manager Williams has built a reputation for developing young talent since taking over in the dugout, and the Jones signing fits a pattern of recruiting players with higher-level pedigree who can fulfil their potential away from clubs where first-team access is more limited.