The deal will pay Wagner $7 million in 2010 and includes a $6.5 million club option for 2011, as well as a $250,000 buyout. The option would become guaranteed if Wagner finishes 50 games next season.
"We feel this is a great start for us to put together a championship-quality team for 2010," Braves general manager Frank Wren said in a statement. "Billy has been at the top of our list as a player we wanted to acquire for some time, and we're excited to have him anchoring our bullpen."
The agreement was first reported by FoxSports.com.
The Boston Red Sox offered salary arbitration to Wagner, so they're entitled to draft-pick compensation when the deal with Atlanta becomes official.
"We knew going into this offseason that we wanted to fix two areas: the bullpen and the offense," Wren said.
Wagner, 38, ranks sixth on baseball's career list with 385 saves, second among left-handers behind John Franco (424). He posted a 1.72 ERA and struck out 26 batters in 15 2/3 innings with the Mets and Red Sox last season after returning from Tommy John surgery.
Calling himself a lifelong Braves fan who grew up watching the team on TBS, Wagner said it's always been his goal to pitch for Atlanta. The deal came together quickly after Wren, manager Bobby Cox and pitching coach Roger McDowell flew to Wagner's rural Virginia home to make their pitch in person.
"I grew up loving Dale Murphy. I remember Bob Horner hitting four home runs in a game," Wagner said. "They have such a great tradition of winning and competing and great guys. I wanted to be a part of that. I'm excited about the opportunity. They'll get the best I have to offer."
Soriano (27 saves) and Gonzalez (10 saves) shared the closing duties last season, but both became free agents and now surely will sign elsewhere. The Braves immediately targeted Wagner, considering him a better option than either of the incumbents.
Wren pointed to Wagner's 86 percent success rate at converting saves over his career.
"Anything over 80 percent is really good," the GM said. "He's got everything you want your closer to be. He's fearless. He's got quality stuff. Even when you consider the job those two guys did for us last year, we had Billy ranked higher."
Wagner said his arm feels better than it has in years.
"I never had any discomfort. I never felt like I had to back off," he said. "If I had known it was going to work out this well, I would have had Tommy John [surgery] five years ago."
The Braves are counting on getting extra picks for Soriano and Gonzalez, who have until Monday to accept arbitration but are expected to turn it down to pursue multiyear deals with other teams.
Atlanta declined to offer arbitration to first baseman Adam LaRoche or outfielder Garret Anderson. There's still a chance that LaRoche might return to Braves, who acquired him from Boston just before the trade deadline for his second stint with the team. He was one of the team's best offensive players down the stretch, hitting .325 with 12 homers and 40 RBIs in 57 games.
"We prefer that it be our decision" on whether to pursue a new contract with LaRoche, Wren said. "By offering arbitration, you're giving that decision over to the player."
If nothing else, the re-signing of Hudson and signing of Wagner shows the Braves are serious about sending Cox out a winner in his final season as manager. Last season, the one-time NL powerhouse made a run at its first playoff berth since 2005 before a dismal final week ended those hopes.
"Guys realize how close we are to winning," Wren said. "That's helping us lure guys here again."
Jerry Crasnick covers baseball for ESPN.com. Information from The Associated Press was used in this story.