After getting roughed up by the first two batters he faced Sunday, the big right-hander settled down and kept Cincinnati rolling.
The Reds also won 10 straight in 1998, and now they are two wins away from tying the franchise mark that has been done twice -- most recently in 1957.
Cincinnati opened a three-game lead over Pittsburgh in NL Central, its biggest edge since mid-June.
"They're not going to go away, that's fairly obvious at this point," outfielder Jay Bruce said of the Pirates. "We have some series left with them, but anytime you can gain a game it's great."
The Reds have swept four of five series since the All-Star break and are a season-best 21 games above .500 (61-40). They have won nine straight road games, including all six on this just-completed trip.
"I'm always talking a perfect road trip, and we finally had one," Reds manager Dusty Baker said.
Latos (9-3) didn't want to be the one to stop the Reds' momentum, and he rose to the challenge. He allowed four hits and struck out eight.
"A lot of pressure coming into the game knowing we won nine in a row, but I just attacked them," Latos said. "Today it was in the back of my mind, but it's baseball. Just play."
"It was a fastball away. It looked like I beat him but he got it up in the air and into a bad part of the ballpark where it flies out," Latos said.
It was the only bright spot for Colorado, which has lost three straight and nine of 11.
"Josh jumped on that first pitch fastball in the bottom of the first inning after Dexter drew a walk, and then that was it," Rockies manager Jim Tracy said.
The Reds took a 3-2 lead into the fifth, then broke it open against Jonathan Sanchez (0-2). Cozart led off with a single, and Stubbs hit his 12th home run.
One out later, Bruce hit his 20th homer into the second deck in right to make it 6-2 and chase Sanchez.
Sanchez, acquired from Kansas City for Jeremy Guthrie on July 20, didn't fare much better than the pitcher he replaced. In his home debut, Sanchez allowed six runs and eight hits, struck out three and walked two. He has failed to pitch at least five innings in either of his starts for Colorado.
"I got hit a lot," he said. "I was throwing strikes, and they were swinging the bats."
Cincinnati took the lead with carbon-copy plays in consecutive innings. In the second, Chris Heisey led off with a triple and scored on Cairo's sacrifice fly. Cozart followed with a leadoff triple in the third and came home on Stubbs' sacrifice fly to give the Reds a 3-2 lead.
"The bats have picked up the past couple of days," Bruce said. "We've been doing what we need to do. Whether it's the long ball, base hits, sac flies, we've been playing really good baseball."
The Reds struck first when Brandon Phillips' single scored Stubbs from second base in the first. The Rockies answered in the bottom half when Rutledge homered.
"Latos threw the ball great," Baker said. "Other than that first-inning, two-run homer he was lights out. We got some timely hits out of guys."
Phillips left the game in the fifth inning with cramping in his left calf but said he should be fine.
"I'm all right. The knot is still there, but I'll be OK," he said. "People were saying it was the altitude, the dry heat, I didn't drink enough water or something. I'm ready to play (Monday). I think I'm going to play."
Heisey, who also doubled, has had multiple extra-base hits in four games this season. ... The Reds went 5-1 against Colorado this season. ... The Reds open a six-game homestand with Mike Leake (4-6) facing San Diego on Monday. ... Colorado's Jeff Francis (3-2) will pitch the first game of a three-game series against St. Louis on Tuesday.