The victory also gave Atlanta a season-high seven-game winning streak.
"He did a terrific job. He mixed his pitches well, his command was outstanding," said Braves manager Fredi Gonzalez. "Even better than I thought he would be his first time out. Facing the New York Mets in the heat and his first time in two years, I think we got every bit of what we wanted out of him."
Sheets (1-0), who was signed to a Minor League contract on July 1, allowed only two hits in earning his first win since July 10, 2010, a 15-1 victory over the Angels as a member of the A's. The 33-year-old right-hander threw 88 pitches Sunday, 57 for strikes, striking out five while walking one.
"I felt pretty good. I've been feeling like I could throw the ball where I want to first and foremost. That's the most important thing," he said. "I was having fun."
The only time it wasn't fun for Sheets was in the third, when catcher Josh Thole's double off the wall in left-center put runners on second and third with two out for David Wright.
Wright came into the game with a National League-leading .368 daytime average and a Major League-leading .382 road average.
After falling behind 2-0, then working the count full, Sheets retired Wright on a fly ball to the warning track in right. The threat was over and the game remained scoreless. The Mets limited Wright to a .182 average in the series (2-for-11) with seven strikeouts, and he was hitless in his last nine at-bats.
"It was a big moment, a big pitch," Sheets said. "One big pitch, that can really turn things around."
That Sheets made that pitch didn't surprise Chipper Jones.
"He's a gamer," said Jones. "He knows in a 0-0 game against [Mets ace Johan] Santana that could possibly be one of the biggest outs in the game. What impressed me was that was the only really stressful inning he had. He really didn't get into trouble after that."
Sheeets didn't allow a baserunner after that, retiring the final 10 Mets he faced.
While Sheets cruised, Santana (6-6) would run into trouble in the fifth.
After deftly navigating through a couple of jams over the first four innings, Santana, who allowed six runs and eight hits, and has allowed 13 runs over his last two starts, covering 9 2/3 innings, allowed the Braves to bat around and plate six runs.
Matt Diaz, who started in left field because of his career .514 batting average against Santana, led off with a double over the head of center fielder Andres Torres. Diaz went 1-for-2 to keep his career average against Santana to .514.
Santana then walked Paul Janish, not getting a couple of close calls from home-plate umpire C.B. Bucknor on 2-2 and 3-2. After Sheets popped out attempting to sacrifice, Bourn dropped a 2-2 pitch into left field for an RBI double that broke the scoreless tie. The hit gave Bourn a five-game hitting streak.
"There's not much you can do. You've got to go back and try to get the next guy or the next pitch," said Santana. "Turns out a couple pitches later, you give up a base hit, a double, and it costs runs."
Mets pitching coach Dan Warthen, who thought Santana had Bourn struck out on 1-2, wasn't as diplomatic. He went to the mound and argued quite demonstratively with Bucknor before getting ejected.
Those pyrotechnics were followed by some by Braves hitters.
Martin Prado hit a sacrifice fly to right and Jason Heyward followed by grounding an RBI single to right. After Chipper Jones laced a single back through the box, Freddie Freeman, who came into the game hitting .714 in the series (5-for-7), hit an 0-1 changeup into the seats in right-center for his 11th homer of the year. It extended his hitting streak to seven.
"We might have caught some breaks on some pitch calls but so did they at some point. We took advantage when we did," said Diaz. "Michael Bourn got that great hit. That just speaks to his character. It was great to see him get that big hit to start the scoring, then Freddie busted it open."
Jones had two hits after seeing his 14-game hitting streak stopped Saturday, while McCann extended his hit streak to seven with a second-inning single.
But the story of the day was Sheets. He dominated Mets hitters, throwing 13 first-pitch strikes to the 21 hitters he faced.
"I like to pitch from in front. That's fun," he said. "I just tried to stay ahead and not get in good hitters' counts."
"You go out there with a game plan and he was getting strike one, which allows him to use his other pitches," said McCann. "He can bury his curveball when he wants to and he was getting ahead with his curveball. When you're throwing multiple pitches for strikes, you can go anywhere you want to go."
Atlanta sees Sheets as a key to getting past Washington, the NL East leader.
"I can't wait to run him back out there again in five days or six days with this off-day," said Gonzalez.
"We are ecstatic," said Jones. "For him to come in and build on our winning streak, seven-game winning streaks don't come along very often. We keep the ball rolling.
"Any time I go through nine innings and I don't get a ground ball, that's a good day. I like those kind of days," he added with a laugh. "So me and him are going to get along just fine."