The 25-year-old front man has joined the Shrimps after scoring 29 goals in 47 appearances for the National League North side last season.
His displays saw him win Stockport’s player of the year award and attract the attention of a number of clubs but the Shrimps acted quickly to secure his signature on a two-year deal.
Port Vale, Hartlepool, Wrexham and Chester were all reportedly interested in the former Crewe Alexandra youngster.
“Jason has had an impressive couple of seasons," said boss Jim Bentley.
"He’s proven to be a top goalscorer and now with full-time football we are confident he will do it at Football League level.
“He’s a good player, very hungry and a focused individual and is eager to be a success.”
Oswell began his career at Gresty Road and had spells with Inverness Caledonian Thistle and Rhyl before making his mark in the League of Wales, scoring 50 goals in 94 games for Newton before moving to Stockport.
He becomes the latest new arrival at the Globe Arena after the Shrimps secured the signatures of former Hartlepool striker Rhys Oates and ex-Grimsby defender Zak Mills.