Elvis Andrus came roaring in from third to snap a 3-3 tie, and Boston never recovered, losing the rubber match of the three-game series.
Beckett declined to talk after the game, as he's done several times in recent weeks.
"The wild pitch, I didn't see it coming," Valentine said. "He really battled them."
And unlike his previous three starts, when multiple runs in the first inning sunk Beckett, he got through the first just fine in this one.
This time, however, Beckett had a hiccup in the fourth inning, giving up three runs. And the wild pitch soured his night for good.
"Josh gave us the zero in the first inning as promised. He pitched a good game," Valentine said.
Beckett (5-9, 4.57 ERA) pitched seven innings, giving up nine hits and four runs while walking two and striking out three.
"He pitched great," said Red Sox second baseman Dustin Pedroia. "He was awesome. The balls we hit they caught. The balls they hit fell in. That's baseball, I guess."
And so it goes for the 49-50 Red Sox, who are 10 1/2 games back in the American League East, and five games off the pace in the Wild Card standings. Valentine's team has lost five of its last six games.
Up next? An off-day in New York on Thursday, followed by a three-game series against the Yankees.
"We definitely need to win the series," said Red Sox first baseman Adrian Gonzalez. "That goes without saying. We need to win two out of three, if not try and sweep them. Game 1 is the most important one, and we'll go from there."
If the Sox are going to win in New York, they had better bring their hitting shoes, which were absent in Texas. Boston scored a total of six runs in the three-game series.
"Any loss is frustrating," Pedroia said. "We've got to score more runs. We've got to just play better. It's frustrating."
Pedroia, who tied the game with a homer in the sixth, had another chance in the top of the eighth. After Pedro Ciriaco singled and stole second, Pedroia laced one to deep left, but David Murphy ranged back and caught the ball at the warning track.
"I was just trying to work away and get him to hit one soft or roll it over," said Rangers reliever Mike Adams. "That last slider got more of the plate than I'd like, but he got himself out and David made a nice play."
The near miss became even more painful in the bottom of the eighth when Nelson Cruz led off the inning with an insurance solo homer against Matt Albers.
Unlike on Tuesday night, Boston couldn't come back against Rangers closer Joe Nathan, going down in order.
The Red Sox jumped out quickly, as Jacoby Ellsbury started the game with a walk and moved to third on a single by Ciriaco. Pedroia got the run home, but not in the fashion he wanted, hitting into a 6-4-3 double play.
In the fourth, the Red Sox extended the lead to 2-0 as Will Middlebrooks -- the pride of Texarkana -- belted a solo homer to left.
"It was a lot of fun," said Middlebrooks, who had a large contingent in the stands from his hometown. "It would have been a lot more fun if we could have won, but it was fun. It was fun to play in front of all my friends and family. I had a decent series, but we lost two out of three. We're here to win games."
Beckett ran into his first jam in the bottom of the fourth. Michael Young started it with a double to right, over the head of Cody Ross. Murphy ripped an RBI double to left-center to make it a 2-1 game. After a single by Yorvit Torrealba pushed runners to the corners, Craig Gentry got the tying run home on a perfectly executed squeeze bunt. The Rangers took the lead when Ian Kinsler's fly ball to right could not be caught by Ross, going for an RBI single.
"The ball just missed Cody's glove with that wind, and it took a little dart on the way down or something," Valentine said. "That was the third run, huh? [Darn] shame."
Pedroia got that run back with his shot to center against Rangers starter Derek Holland. But it wasn't enough.
"We're going to play as hard as we can the rest of the way," Pedroia said. "It's not like we're quitting. We're going to go out and play our tails off."