The Braves appeared to be at a distinct disadvantage with Minor, filling in with Tommy Hanson on the DL, going against the two-time NL Cy Young Award winner.
But Minor (3-2) pitched brilliantly, allowing just four hits and one intentional walk while striking out a season-high nine. Only two runners got as far as second base.
Lincecum (11-10) was no slouch, surrendering five hits in seven innings with seven Ks. But the wiry right-hander made one big mistake, serving up a high fastball to Jones leading off the fifth. He sent a drive soaring into the right-field seats for 11th homer, giving the Braves all the offense they would need.
Eric O'Flaherty, Jonny Venters and Craig Kimbrel closed out the five-hitter with one scoreless inning apiece. Kimbrel struck out two in a perfect ninth for his 37th save in 42 chances, breaking a tie with Todd Worrell for the most saves by an NL rookie since 1969.
Worrell had 36 saves for the St. Louis Cardinals in 1986. Kimbrel is closing in on the major league mark for rookies, set just last season by Texas closer Neftali Feliz with 40.
Kimbrel has not allowed a run in his last 28 appearances, covering 28 2-3 innings. Over that span, he's given up a mere nine hits and eight walks, with a staggering 48 strikeouts.
The four-game set was similar in style to the thrilling NL division series between the teams last fall. San Francisco knocked off the Braves 3-1 in the playoffs, with every game decided by one run. This time, Atlanta took three of four, the games decided by a total of five runs.
The Braves wasted a chance against Lincecum in the fourth. Michael Bourn led off with a single and raced around to third on Brian McCann's one-out double into the right-field corner. But Dan Uggla struck out swinging and Freddie Freeman grounded out.
Freeman came up with a brilliant defensive play in the fourth, sprawling out on the rolled-up tarp to make a great diving catch on Aubrey Huff's foul popup.
The Braves won six of seven against San Francisco this season. Atlanta's latest win stretched its lead in the NL wild-card race to six games over the Giants, who came in 2 1/2 games behind Arizona in the NL West.
NOTES: Braves RHP Peter Moylan will begin a rehab stint at Triple-A Gwinnett on Monday and likely will rejoin the Braves when the rosters expand Sept. 1. ... San Francisco was without closer Brian Wilson for the third night in a row. He's bothered by a sore elbow, though the Giants are hopeful he won't have to go on the disabled list. ... RHP Derek Lowe will start for the Braves in the opener of a three-game series against Arizona on Friday. Lowe is 6-9 with a 4.14 ERA in 19 career starts against the Diamondbacks. ... The Braves lost two previous meetings at Arizona in May. ... The Giants continue their road trip with their first three games of the season against Houston. ... Ryan Vogelsong will try to improve to 3-0 in his career against the Astros, who will be in San Francisco next week for a four-game series.
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