After one season in Vegas, during which he helped lead the expansion Golden Knights to the Stanley Cup Final, Neal gets a five-year deal with an average annual value of $5.75 million.
The Golden Knights took Neal from Nashville in the expansion draft last summer. He was expected to be a trade piece at the deadline, but Vegas' unprecedented success changed that.
Neal, 30, scored 44 points in 71 regular-season games and became an emotional leader of a Knights team that surprised the hockey world. He has now lost in the Final in back-to-back seasons.
Neal is coming off a five-year, $30 million contract.
Vegas has now lost two left wingers, Neal and David Perron, who signed with St. Louis.
The Flames also signed former Carolina forward Derek Ryan, giving him a three-year deal with an average annual value of $3.125 million. Ryan, 31, had a career year with 38 points in 80 games last season.
Calgary finished fifth in the Pacific Division last season and missed the playoffs.