Weeks dropped a double into left field and later scored on a sacrifice fly by Josh Reddick. The A's held on to that 1-0 lead until the fourth inning, when the Rangers batted around and scored four runs.
Josh Hamilton led off the fourth by extending his hitting streak to 16 games with an infield single just past A's pitcher Tommy Milone, which is the longest in the American League and the third longest in the Majors.
Then Adrian Beltre hit a first-pitch changeup from Milone into the right-field seats, giving the Rangers a 2-1 lead.
"We needed this one," Beltre said. "The last two games we played before this, we didn't play well. Defensively, we threw the game away, and offensively, we didn't get something going. It was nice to get back to playing our game."
Nelson Cruz became the third straight batter to reach base when he hit a ground ball to shortstop Eric Sogard, who overthrew first baseman Daric Barton. One out later, Yorvit Torrealba walked, and one out after that, Craig Gentry beat out a routine ground ball to A's third baseman Josh Donaldson, allowing Cruz to score and extending the Rangers' lead to 3-1.
"I didn't think there was any way I would beat it out," said Gentry. "I just hustled down the line hoping something would happen, and it did."
It was Gentry's second hit of the game, with his first being a bunt single in the third inning, and it proved to be crucial. The two-out RBI single extended the inning and allowed the Rangers to score again before the fourth ended.
"That was a big infield hit," said Washington. "I don't know how they could have forgotten he could run like that, but that's not where you can play when Gentry puts it in play. That got us a run, and then Elvis [Andrus] came up and got us another one. We played hard tonight, and it paid off for us."
Andrus followed Gentry's infield single -- the Rangers' third of the inning -- with a single of his own, a base hit to left field that drove home Torrealba and gave the Rangers a 4-1 lead. Andrus led off for Texas for the second time this season, moving up from the second spot in the order after Ian Kinsler was scratched from the lineup just before the game with a stomach virus.
The Rangers were also without Michael Young on Wednesday. Young, who has batted .143 with one RBI in his past eight games, was given a day off as Washington has rested many of his regulars this week. Andrus and Cruz sat out of Tuesday's game against the Royals, while Mike Napoli and Kinsler did not play on Monday night.
In his previous outing, Darvish allowed three hits and three runs while striking out seven in 5 1/3 innings, having to deal with a one-hour, 56-minute rain delay, in a 10-3 win over the Angels last Friday night. Darvish said there were no lingering effects from that outing.
"I was able to rest my body and in between [outings]. I was able to rest and prepare normally," Darvish said. "Today, I think the good part was that I was able to throw with less effort. I felt really good."
Darvish improved to 6-1 on the season, breaking a tie with Milone, who fell to 5-3, for most wins by a rookie this year.
"I didn't think his stuff was super electric, but he located the ball really well and kept us off balance with that big curveball that he has, and then he pumps it up to 94 [mph]," Donaldson said. "I really feel like we could have hit him tonight. We just didn't take advantage of some pitches that were left over the plate."
Darvish threw 73 of his 118 pitches for strikes and lowered his ERA from 2.84 to 2.60. Darvish has struck out 58 batters this season, third most in the American League behind only the Mariners' Felix Hernandez and the Yankees' CC Sabathia.
"He went out and did a good job for us," Washington said. "He's known as a strikeout pitcher, and he's getting strikeouts. He has some tremendous stuff. I think each time that he takes the ball, there's something new you might see."
Darvish is 4-0 with a 3.00 ERA in four home starts this season, and has now won each of his three starts against AL West teams.
Joe Nathan struck out the side in the ninth inning to pick up his eighth save and his first since a 5-2 victory against the Indians on May 5.
Professional golfer Phil Mickelson and actor Gary Sinise, who threw out the first pitch, were among the 46,370 in attendance at the Ballpark in Arlington on Wednesday night, marking the 12th sellout for the Rangers this season.