Arenado hit a three-run homer, his 25th of the season, in the first inning, when the Rockies jumped on Braves starter Tyrell Jenkins for four runs.
Story belted his 27th homer in the fourth, a two-run shot that made it 7-0. It was Story's fourth homer in three games and fifth in his past seven. During that span, Story has driven in 12 runs, giving him 69 RBIs for the season.
Story, who has homered in three consecutive games, one short of his career high, leads the National League in home runs. Arenado, who topped the NL last year with 42 homers, is tied for second with the Chicago Cubs' Kris Bryant.
DJ LeMahieu, who had three hits, led off the fourth with his career-high-tying sixth home run, matching his 2015 total.
Rockies starter Tyler Chatwood had a bizarre outing. He allowed one hit -- a first-inning single -- in five scoreless innings. However, Chatwood issued a career-high eight walks -- the most by a pitcher in the majors this season -- and struck out six.
Chatwood (9-6) never previously walked more than six. He threw just 51 strikes among his 97 pitches Sunday.
"Effectively wild, I think they call that," said Chatwood, who won for the first time in five starts since June 11. "I thought my stuff was really good today. I thought the ball was coming out good. I had good life to it.
"It's weird. Especially in this ballpark, normally if you walk guys, you're going to get hurt here."
Chatwood gave up two walks and a single in the first but escaped. He issued two more walks in the second, but sandwiched between them was a hit-robbing catch by shortstop Story of A.J. Pierzynski's flare to short center. Chatwood then struck out Jenkins and Jace Peterson to end the inning.
"It seemed like we should've been the one running away with the game, really, with all the baserunners we had," Braves interim manager Brian Snitker said. "We just couldn't pierce a gap or get a hit or get anything going and take advantage of all the walks."
The sweep of the Braves enabled the Rockies to finish their homestand 5-2, improve to 7-3 in the second half go over .500 (25-24) at Coors Field for the first time since they were 4-3 there April 22.
Held scoreless until their two-run seventh, the Braves were swept in a four-game series for the second time this year. They are 1-6 on their current 10-game road trip, and they were outscored 26-12 in their four losses to the Rockies.
"I think that everybody's putting the work in and doing things," Atlanta left fielder Jeff Francoeur said. "It's just there's no results. And that's tough, especially when you get beat the way we got beat really in three out of four of these games."
The Rockies have won 10 straight meetings with the Braves at Coors Field, matching their record win streak there against any team. Colorado also won 10 consecutive home games against the Cincinnati Reds from Aug. 23, 2008-Sept. 9, 2010.
The Rockies have won 14 of their past 16 games overall against the Braves and swept them in a four-game series for the second time in two seasons.
Arenado homered for the second time in three games after Charlie Blackmon led off the first inning with a single and LeMahieu walked. Arenado connected on a first-pitch fastball from Jenkins.
"Just didn't execute pitches at first," Jenkins said. "Got those two guys on early, and that pitch to Arenado was up, and he made me pay for it."
Story doubled with one out and scored on Nick Hundley's two-out single in Jenkins' 37-pitch inning that included just 19 strikes.
In his third career start, Jenkins (0-2) allowed eight hits, including six for extra bases, and seven runs with five walks and two strikeouts. He threw 87 pitches, 45 strikes, in 3 1/3 innings.
"He's just got to be more aggressive," Snitker said. "It was almost like he was kind of feeling for the strike zone a little bit instead of on the attack and turning the ball loose. He's had two pretty good starts and just didn't have it today. This place -- if you don't have it here, it's going to let you know in a hurry."
NOTES: SS Trevor Story, 23, on Saturday became the youngest player to have a 4-for-4 or better game with two homers and one stolen base since 21-year-old Joe Morgan went 6-for-6 in 1965, according to the Elias Sports Bureau. ... Braves RHP Julio Teheran felt fine after playing catch and expects to make his next scheduled start Thursday against Philadelphia. He came out of his Friday start in the fifth inning due to right lat tightness. ... Beginning Monday, the Rockies play 29 of their next 35 games against teams that are at least seven games above .500. ... Rockies LHP Chris Rusin (left shoulder strain) made his third rehab start Sunday night for Triple-A Albuquerque. He will join the Rockies in Baltimore, where they begin a series Monday, and be re-evaluated.