Tommy La Stella also had three hits and Chris Johnson drove in two runs as the Braves moved to within 1 1/2 games of NL East-leading Washington, which lost 1-0 to Cincinnati.
Teheran (10-6) allowed three runs on nine hits and no walks in six innings. He had five strikeouts while becoming the first Braves pitcher to reach 10 wins.
Despaigne (2-2) walked five batters and also was wild on a crucial throw to third base for an error in Atlanta’s four-run fourth inning. The right-hander from Cuba lasted only 3 2/3 innings and allowed five runs, two earned, on six hits.
Seth Smith had three hits and Yasmani Grandal had a two-run double for the Padres.
Craig Kimbrel struck out the side in the ninth for his 31st save.
Despaigne made his first start since throwing 7 2/3 no-hit innings before allowing a run in a no-decision against the Mets on July 20.
Teheran drove in a run with a squeeze bunt in the fourth.
Despaigne walked three batters in the fourth and hurt himself with a high throw to third base for an error when trying for a force play on Chris Johnson.
San Diego third baseman Chris Nelson fell over the sliding Johnson after reaching for the high throw. Johnson then scored on Teheran’s bunt.
Johnson and Freeman had run-scoring singles in the inning.
The Padres took a 2-1 lead off Teheran in the fourth. Grandal’s double to centerfield drove in Yangervis Solarte and Smith, who reached on singles. Solarte had a run-scoring single in the fifth.
Atlanta scored one run in the second when Jason Heyward walked, moved to third on a double by Evan Gattis and scored on Johnson’s groundout to third base.
Padres first baseman Yonder Alonso had one hit in his return from the 15-day disabled list. Alonso had been out since June 17 with right wrist tendinitis.
Braves manager Fredi Gonzalez had an unsuccessful challenge of a foul call on Freeman’s liner down the left-field line in the fourth. Freeman had a run-scoring single up the middle after the ruling that the play stood.
The Padres committed three errors.
NOTES: The Braves will show Sunday’s Hall of Fame induction ceremony on the Turner Field video board. Gates will open at 1 p.m. for the 5:05 p.m. game, and fans with tickets to the game can watch the ceremony. Former Braves pitchers Tom Glavine and Greg Maddux, former manager Bobby Cox and former player and manager Joe Torre are among the inductees. ... Padres OF Jake Goebbert was optioned to Triple-A El Paso. ... Braves 2B Phil Gosselin reached on an error in the sixth and was picked off base following his recall from Triple-A Gwinnett. IF Tyler Pastornicky was optioned to Gwinnett. ... The series continues Sunday when Atlanta’s Mike Minor faces San Diego’s Eric Stults in a matchup of left-handers.