The team announced the deal Tuesday.
Vesey has 33 goals and 22 assists in 159 games in the past two seasons.
He had 17 goals and 11 assists in 79 games last season. He tied for the team lead in game-winning goals (four) and was third in even-strength goals (16).
Vesey, who won the Hobey Baker Award as the top player in collegiate hockey in 2015-16, was signed out of Harvard by the Rangers as a free agent on Aug. 19, 2016. He was originally selected by Nashville in the third round of the 2012 draft.