The deal includes $4.5 million more in potential performance and roster bonuses, the source said.
The deal does not include a club option for 2015, meaning Floyd will be able to go back on the market a year from now as a free agent.
Floyd, 30, is 70-70 with a 4.48 ERA over parts of 10 seasons with the Philadelphia Phillies and Chicago White Sox. His best season was 2008, when he went 17-8 with a 3.84 ERA in 206 1/3 innings with Chicago.
He underwent Tommy John surgery to repair a torn ulnar collateral ligament in his right elbow in May and is hoping to be back in the big leagues by May 2014, barring setbacks in his rehab.
Floyd said his arm "feels tremendous."
"It's nice to start over, create the right mechanics ... all the simple things to become more natural," he said.
The Atlanta Journal-Constitution earlier reported that the Braves were nearing an agreement with Floyd.
Floyd said he liked the Braves, who won the NL East in 2013, for their consistent success.
"It just seemed like all the arrows were pointing to Atlanta," he said.
The Braves are among the teams pursuing Tampa Bay Rays starter David Price and Chicago Cubs pitcher Jeff Samardzija in trade, but general manager Frank Wren said last week that he's hesitant to dig too deeply into the farm system to add a top-of-the-rotation starter.
Pitchers Lucas Sims, J.R. Graham and Jason Hursh and catcher Christian Bethancourt are Atlanta's most highly regarded prospects.
"Trading players you have for a long time for one year or two years of service [from a player] doesn't work and doesn't make sense," Wren told ESPN.com on Thursday from the winter meetings in Florida. "We're in a good spot. We like our five guys. They're all homegrown, and they're all on the ascent. We think we can go into the season with our rotation as it is and have a very solid group."
The Braves also announced Monday they avoided arbitration with Beachy by reaching an agreement on a one-year deal.
Information from The Associated Press was used in this report.