Wright was not offered a new contract for the 2022 season as the Cowboys confirmed several list changes last month.
Wright will now venture abroad to link up with Salford, who managed an 11th placed finish this season after recording seven wins from their 22 matches.
“It’s a new chapter in my rugby league career and I’m excited to come over here and get stuck in," Wright said on the announcement.
“I’m looking forward to improving my game and buying into the culture at the club."
A Perth product, Wright moved to the east coast in 2015 to play with the Titans' NYC side before signing with the Cowboys the following year.
Red Devils director of rugby Ian Blease stated his excitement to have Wright join the Greater Manchester side for the next two seasons.
“I’m delighted that we have landed Shane’s signature," Blease said.
"Shane is bringing with him four years of valuable NRL experience and after speaking with Shane and watching him as a consistent performer at the Cowboys last season, he is coming to the Red Devils bringing his workmanlike attitude, strength and agility to the pack for next season and beyond.”
The 2022 Super League season is set to commence in February, consisting of 27 rounds in the lead up to the 2022 World Cup.