Instead, it was Lincecum who cracked.
"He's obviously the better pitcher," Minor said. "It's a big win."
The left-hander 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 against Minor (3-2), who didn't flinch when Cody Ross led off the game with a double.
"Instead of giving in, I tried to bear down," Minor said. "I knew if he scored, that would be a big run."
Lincecum (11-10) was no slouch, surrendering five hits in seven innings with seven Ks. But the wiry right-hander made one pitch he wanted back, 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.
"You know you're not going to hit too many homers off Lincecum," Jones said. "He made one mistake, and that's all we needed."
Lincecum was trying to go inside with the pitch, but it faded back over the plate.
"Chipper's a great hitter," the Giants starter said. "I gave up that one timely home run. That cost me."
Eric O'Flaherty, Jonny Venters and Craig Kimbrel closed out the five-hitter with one 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.
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.
"I'm glad I don't have to face anybody in our bullpen," Jones said.
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 Giants had to use a makeshift lineup because of injuries, while the Braves showed off some of their impressive young pitching talent. Randall Delgado threw six-plus innings of one-hit ball on Tuesday before heading back to the minors. Another rookie, Arodys Vizcaino, picked up his first big league win in that game. Minor finished the series with another strong effort.
"It's almost embarrassing how many young pitchers who can help this club we have at the minor league level," Jones said.
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 2½ games behind Arizona in the NL West.
"We might face these guys later on," Jones said, referring to the playoffs. "Every mental edge we can get is helpful."
Atlanta 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.
"We've just got to find our luck," Lincecum said. "We've got to find it soon."
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.