The Braves lead the NL East by 10 games over Washington. Atlanta opens a three-game series at Nationals Park on Monday night, and can clinch the division by winning twice there.
Smith (1-1) held the Braves hitless until pitcher Julio Teheran singled to lead off the sixth.
Smith allowed three hits and walked two. The rookie made his fifth career start - he had a 12.41 ERA in his previous starts. He struck out seven of the first 13 Atlanta batters.
Teheran (12-8) gave up seven hits, four runs, no walks and struck out six in 6 2-3 innings. Manager Fredi Gonzalez pulled him with two outs in the seventh after Burch bunted into a forceout.
Jedd Groyko's RBI single and Headley's 13th homer in the sixth gave the Padres a 3-0 lead.
Medica, who homered off Philadelphia's Cliff Lee last Wednesday for his first career hit, led off the seventh with his second homer to make it 4-0.
After Teheran left, reliever Scott Downs gave up a double to Will Venable that advanced Burch to third. After pinch-hitter Chris Denorfia walked to load the bases, Downs struck out Gyorko to end the threat.
Smith retired the first 11 batters before Freddie Freeman walked in the fourth. Freeman moved to second when Smith made a wild pickoff throw, but Evan Gattis struck out.
Padres closer Huston Street pitched the ninth. He gave up a leadoff single to Freeman before retiring the next three batters.
Teheran had retired 14 straight batters before Venable singled to begin the sixth. One out later, Venable stole second and scored on Gyorko's single.
NOTES: In the first six games of San Diego's 10-game road trip, Headley is hitting .364 with three homers and seven RBIs in 22 at-bats. ... Venable, who went 3 for 4, became the 10th San Diego player to have a 20 homer-20 steal season. ... Gattis went 0 for 4 and is hitless in his last 15 at-bats. ... Braves CF B.J. Upton, who went 0 for 3 and struck out twice, is 1 for his last 21. His average dropped one point to .190. Upton has struck out 143 times in 378 at-bats.