5. 1961 Milwaukee Braves - 35.6 WAR
You are going to sense a familiar theme with this list as any offense that has Hank Aaron on it is generally going to be pretty good. The 1961 Milwaukee Braves finished the season 83-71-1 and was loaded with offensive studs. If they had some better pitching other than Warren Spahn who put up a 3.02 ERA in 262.2 innings, they could have really done some damage.
Leading the offense predictably was Hank who posted a 9.4 rWAR season, but Eddie Mathews held his own with 7.3 rWAR as well. Frank Bolling chipped in with 4.2 rWAR and even a very young Joe Torre made a nice contribution with 3.4 rWAR. All in all, just a very solid offensive team led by two of the best hitters to ever wear a Braves uniform.
4. 2003 Atlanta Braves - 36.3 WAR
Jumping ahead several decades, we have the 2003 Braves offense with 36.3 rWAR for the season. This team won 101 games and won the National League East, hit 235 home runs (a franchise record at the time), but ultimately lost in the the NLDS to the Cubs 3-2. If you ever want to know one of the several reasons why Braves fans don't remember Mike Hampton fondly, you should go look at that series.
This was a fun offense that included five players that made the All-Star team in Marcus Giles, Javy Lopez, Gary Sheffield, Rafael Furcal, and Andruw Jones and that is also the same order they finished in WAR with Giles' 7.9 leading the way. Of note, Chipper Jones did exist, but he didn't make the All-Star team that year. However, he did still put up 4.4 rWAR in 2003 and obviously things turned out well for him and the Braves overall.
3. 1964 Milwaukee Braves - 36.9 WAR
Heading back in time again, the 1964 Milwaukee Braves take home the bronze here with 36.9 rWAR. They went 88-74 in their second to last season in Milwaukee and again, this team was loaded. Unfortunately, pitching proved again to be a problem as Warren Spahn was at the end of this Hall of Fame career, Tony Cloninger was the only starter that really put up decent results, and Phil Niekro was just a rookie that didn't even make a start that season.
Hank Aaron led the way with 6.8 rWAR and was followed very closely by Denis Menke's 6.7 WAR season. This was a very well-balanced lineup as Joe Torre, Rico Carty, and Eddie Mathews all put up 4.6 WAR or better for the season.