The Braves announced Thursday that Smith signed a three-year, $39 million deal with the team. The agreement also includes a 2023 club option for an additional $13 million.
League sources told ESPN the contract includes a buyout that will pay Smith $1 million if the Braves decide to not pick up the club option.
"I'd like to thank the Giants for giving me the opportunity that they did back in 2016," Smith said in a statement. "It was my first taste of the playoffs I've had in my career and it is something I'll never forget. The Giants are a first class organization hands down and welcomed me and my family with the utmost respect.
"Without their training staff, coming back from [Tommy John] surgery would not have been possible and for that I am forever thankful. ... Thank you for all the good times. And to the fans, thank you for the constant support and loyalty."
Smith, 30, recorded 34 saves in 38 opportunities for the Giants this past season. He had a 6-0 record with a 2.76 ERA. The Georgia native struck out 96 while walking 21 and was named to the All-Star team for the first time in his MLB career.
Smith previously received a $17.8 million qualifying offer from the Giants. He signed with the Braves shortly before the deadline to accept that offer. Atlanta will forfeit its second-highest pick in the 2020 draft for signing the veteran closer.