Also, the Rangers announced right before the game that they had acquired right-hander Matt Garza from the Chicago Cubs.
"Last few years when we get pitchers like him we go to the World Series," said Nelson Cruz, who homered. "So that's good news."
Darvish (9-4), who last pitched July 6 before going on the disabled list with a right trapezius strain, struck out four and walked two. He left after allowing consecutive batters to reach in the seventh, having thrown 60 of 90 pitches for strikes.
The Yankees could have to face the newest Texas ace before they leave town. Garza, pulled from his scheduled start for the Cubs at Arizona on Monday night, is likely to make his Texas debut on Wednesday night.
"You've got to commend (general manager) Jon Daniels and the group the way it worked to try to put this together, and they got us what we needed," Washington said. "We got a tremendous starting pitcher."
The Rangers had lost seven of eight games, getting swept at home in a three-game series by Baltimore over the weekend after the All-Star break.
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Milwaukee's Ryan Braun, the 2011 National League MVP, was suspended without pay for the rest of the season and the postseason Monday, the start of sanctions involving players reportedly tied to a Florida clinic accused of distributing performance-enhancing drugs. Cruz is among more than a dozen other players also targeted by MLB for having connections to the now-closed anti-aging clinic in Miami.
"I guess it is what it is. I don't have any comment," Cruz said when asked about Braun.
Cruz also said he's "playing the same way. We're playing to win. We're in the race so I come to the ballpark thinking about winning. So that's my goal."
Three relievers wrapped up the three-hitter that was the eighth shutout of the season for the Rangers. All-Star closer Joe Nathan pitched the ninth for his 31st save in 32 chances
Texas had reliever Robbie Ross warming up while batting in the sixth. But Darvish came out to start the seventh, and appeared surprised when Washington came out of the dugout to make the pitching change.
"He was a little bit earlier than I thought, but I had about 15 days with a break and I think Wash was thinking about my body and how I felt, so I really appreciated what he's done," Darvish said through an interpreter.
New York didn't have a hit until Lyle Overbay led off the fifth with a bloop single into short left field that was out of the reach of third baseman Adrian Beltre. The Yankees didn't have another hit until Overbay lined a single to right in the seventh, just before Vernon Wells walked and Darvish came out of the game.
"It seemed like he located really well," Yankees manager Joe Girardi said. "His fastball, our guys were talking about it, he was giving us first-pitch fastball, but it was all on the corner or down in the zone, he wasn't giving them a whole lot to swing at."
Ivan Nova (4-3) struck out four while throwing a season-high 112 pitches over seven innings.
The Yankees, coming off an 11-inning loss late Sunday night at Boston, were held scoreless for the eighth time.
"It's something we have to deal with, so we got to find a way to do that," Overbay said. "Tonight, I don't know if any of those had an effect on us. It was more Darvish was on his game."
Cruz's 23rd homer was a 412-foot solo shot in the seventh that landed in the Yankees' bullpen in left-center field and made it 3-0.
The Rangers led for good after Ian Kinsler drew a leadoff walk in the first, stole second base and scored on Beltre's single -- the first run they've scored in the first inning in 11 games. In the sixth, Elvis Andrus had a walk and a stolen base before Geovany Soto doubled.
Darvish only missed one start for Texas, but was unable to pitch in the All-Star Game even though he did go to New York last week. ... Texas sent rookie right-hander Justin Grimm, top prospect third baseman Mike Olt and Class-A pitcher C.J. Edwards to the Cubs. There will also be at least one player to be named later, possibly two, depending on who is chosen by Chicago. ... The Rangers' bullpen has pitched 22 1/3 consecutive scoreless innings. ... Yankees OF Curtis Granderson, expected back early next month from a broken pinkie, got three at-bats in his first simulated game in Tampa and continued running the bases. Granderson said he feels like it's "at the spring training point again" where preseason games are just a few days away. His status will be evaluated after simulated games Tuesday and Wednesday.