The victory came one night after the Rangers had endured a 17-5 loss to the Red Sox on Tuesday.
"We'll take it, especially after the tough game yesterday," Nathan said. "It was a good win for us, and now we have a chance to come back tomorrow and win the series."
After Ogando and Red Sox starter John Lackey had pitched to a 1-1 standoff through six innings, Elvis Andrus delivered the big blow with a two-run double off reliever Koji Uehara in the top of the seventh. The Rangers' only other run came on a fourth-inning home run from Adrian Beltre.
"This was a huge win for us," Andrus said. "We have been really good at turning the page and getting ready for the next game. That's the way we've been the last few years, and [we] were tonight."
Ogando, who was activated off the disabled list before the game, was terrific for 5 2/3 innings, at least until his final pitch of the night. He took a two-hit shutout and a 1-0 lead into the sixth before Dustin Pedroia lifted a two-out fly ball over the left-field wall for a game-tying home run.
That came on Ogando's 92nd and last pitch of the game. But manager Ron Washington loved what he saw from Ogando in his first outing since May 15. He had been sidelined with triceps tendinitis and had made just one 60-pitch rehab start before facing the highest scoring offense in the Major Leagues.
"They've got a lot of threats over there, so you have to give Ogando a lot of credit to go through a lineup like that," Washington said. "We needed exactly what he gave us. He was on top of his game. He was sharp and able to execute pitches. If you don't execute pitches, they'll make you pay."
Ogando was especially sharp in a crucial moment in the fourth inning. He was looking at runners at first and second with one out, but struck out both David Ortiz and Mike Napoli to end the threat.
"That was a great moment," Ogando said. "That was a big inning. I had to focus on every pitch. I had to concentrate and not throw any mistake pitches."
The home run to Pedroia cost him the victory. That went to Cotts, who took over after the home run and walked both Ortiz and Napoli. But he struck out Jarrod Saltalamacchia to end the inning. That put the reliever in position to get the victory after Andrus put the Rangers ahead with his two-run double in the seventh.
It was Andrus' first extra-base hit since May 18, as he jumped on a first-pitch fastball from Uehara. It was also the only hit either team had with runners in scoring position on the night.
"I think it was just the location that I missed," Uehara said through an interpreter. "Strategy-wise, I think I was perfectly fine."
Cotts left after a leadoff walk to Stephen Drew in the seventh. That meant he walked three of four batters, but still got his first victory since July 1, 2006, while pitching for the White Sox against the Cubs.
"What a way to get it, huh?" Cotts said. "I'm just happy the team got a win and the rest of the bullpen held it down."
It wasn't easy. Robbie Ross gave up a single to Jose Iglesias that put runners on first and second. But Ross struck out Jackie Bradley, got Daniel Nava on a weak grounder back to the mound that advanced the runners, and then got Mike Carp on a weak fly to center to end the inning.
In the eighth, Tanner Scheppers allowed a two-out walk to Napoli and a broken-bat double to Saltalamacchia. That scored Napoli and made it a one-run game. But Scheppers retired Drew on a grounder to second to end the inning.
The Red Sox were 0-for-7 with runners in scoring position. Only Nathan spared the Rangers any drama. He retired the side in order in the ninth on just six pitches for his 18th save. He is also 16-for-16 against the Red Sox in his career, including 7-for-7 at Fenway.
"That's what this ballpark does, it's a tough place to get three outs," Nathan said. "I have been part of some games where it has been very dicey, so to have a six-pitch inning, I'll take it."