Alex Cobb scattered three hits, walked one and struck out six in seven shutout innings for the Rays.
"Unfortunately for us, he was really on," Cobb said of Darvish.
Only one runner reached second against Cobb.
"He didn't give up any pitches," Andrus said. "I thought he was painting the whole game. I'm really glad that we had Yu out there. He was doing the same thing for us."
Tampa Bay had runners on second and third with one out in the fifth, but failed to score when Wil Myers hit an infield pop fly and Ben Zobrist flew out.
Darvish got three consecutive outs after allowing Evan Longoria's double to lead off the sixth. He also worked out of a jam during the seventh after Ryan Hanigan hit a leadoff double.
"We just did not get the knocks when we needed it," Tampa Bay manager Joe Maddon said.
The Rays were hitless in 10 at-bats with runners in scoring position against Darvish, and 0 for 11 overall.
"As usual, you've got a team over there that fights forever," Washington said. "They put themselves in some good positions to score some runs, but Yu made the pitches when he had to."
Darvish was activated from the 15-day disabled list and RHP Nick Martinez was optioned to Double-A Frisco. ... Rangers RHP Colby Lewis (hip surgery) gave up four runs and four hits over five innings in a rehab start for Triple-A Round Rock. ... Texas LHP Joe Saunders (bruised left ankle) is not expected to be ready for his regular bullpen session Monday and could miss his next start. ... Texas RHP Tanner Scheppers (0-0) will face Boston RHP John Lackey (1-0) Monday night in the opener of a three-games series at Fenway Park. ... Tampa Bay LHP Matt Moore (0-1) and Royals LHP Jason Vargas (0-0) are the scheduled starters for Monday night's game in Kansas City.
Information from The Associated Press was used in this report.
Copyright by STATS LLC and The Associated Press
ST. PETERSBURG, FL - APRIL 6: Alex Cobb #53 of the Tampa Bay Rays pitches during the first inning of a game...
Yu Darvish held right-handed hitters to just a .500 OPS on the road in 2013 while striking out batters in 43.1 percent of their plate appearances (second in MLB). At home in 2013, Evan Longoria struck out in 23.3 percent of his plate appearances against right-handed pitching, which was sixth worst in the American League.