Jones hit his ninth homer -- a three-run shot off Bronson Arroyo in the seven-run second. Kelly Johnson was 4-for-6 with three doubles, and Mark Kotsay was 3-for-4 with a homer.
The Braves outscored the Reds 25-8 in the three games. Cincinnati has lost five in a row.
"We caught a team when they were down," said Jones, who was 3-for-6 and is hitting .425.
Like the Reds, the Braves have suffered in away games. They're 11-4 at home but 4-11 on the road.
Jones said the Braves, who returned to .500, should use the series against the Reds as a template for the rest of their season.
"When we learn to play on the road like we did, the same aggressiveness, step on somebody's throat when we get them down, when we learn to do that ... I think wins on the road will come," Jones said.
The Braves, who matched their season high with 15 hits in Saturday's 9-1 win, set a new high with 19 on Sunday.
"We've got to keep it going," Johnson said. "We have the potential to have a lot of days like this. We've got to save some runs for those one-run games we've been losing."
The Braves are 0-9 in one-run games.
Arroyo (1-4) lasted only 1Q innings, the shortest of his 166 career starts. His ERA climbed to 8.63, as he gave up seven runs.
Tom Glavine couldn't take advantage of an excellent chance for his first win of the season. He gave up seven hits with five walks and six runs in 4w innings.
"Too bad," manager Bobby Cox said. "We got Tommy some runs, finally, and he didn't have his best day."
Glavine was one out from qualifying for the win when he was lifted with the bases loaded.