A crowd of 30,264 saw the Rangers lose their eighth straight to the Red Sox this season and ninth straight going back to last season. Boston has won 26 of the past 36 between the two teams.
"Obviously, we haven't played our best ball against them, but a lot of that has to do with them," Millwood said.
The Rangers have been outscored 75-36 in eight games against the Red Sox this year.
"Tonight was just an ugly game," Rangers shortstop Michael Young said. "We had a couple of defensive miscues and we didn't swing it well. Obviously, Beckett threw the ball very well. He located his pitches and couldn't get anything going."
The crowd included a large contingent of Red Sox fans that made themselves heard for much of the night.
But as manager Ron Washington said, "If we could have done some things, we could have shut them out."
The Rangers didn't. Instead, the crowd saw Millwood have his three-game winning streak snapped while Texas was almost shutout for sixth time in 46 games since the All-Star break. Nelson Cruz kept that from happening with a two-out RBI single in the ninth, but the Rangers have scored just two runs in their past 19 innings.
"Getting five innings from Beckett was great, but you still have to pitch well against a team with the best offense in the league," Red Sox manager Terry Francona said. "Our bullpen did a good job having to go four innings tonight."
Millwood, despite allowing a second-inning home run to Mike Lowell, looked strong through 3 1/3 innings. Then second baseman Joaquin Arias bobbled a routine one-out grounder by Mark Kotsay and the night started unraveling.
Jason Bay lined a sharp double to left and Lowell singled to left to drove home both runners. That gave Boston a 3-0 lead, and the Red Sox then loaded the bases with a walk to Jed Lowrie and a single by Jason Varitek.
Coco Crisp followed with a short pop to center. Arias raced out and got his glove on the ball but couldn't make a tough back-handed catch with his back to plate. If he made the catch, Millwood might have been able to get out of the inning. Instead, Arias couldn't hold it. The play was scored a single and two runs ended up scoring, giving the Red Sox a 5-0 lead.
"It would have been nice to tighten up the defense because Millwood was throwing the ball well," Young said. "The home run was a solo shot, that's no big deal. But if we could have tightened up and played good defense, I think it would have been a matter of time before we had a big inning. But neither happened."
Better defense would have helped, but Millwood was still willing to shoulder some of the blame despite the error by Arias.
"One out, a runner on first, you should be able to do something with that," Millwood said. "You're going to make errors, I've got to do a better job of trying to get us out of it."
The Red Sox made it 7-0 in the sixth inning on a two-out single by Varitek and back-to-back triples by Crisp and Jacoby Ellsbury. By that time Beckett had things well in hand.
Beckett allowed just four singles in five innings. He did not walk a batter and struck out seven.
"Every time I face Josh Beckett, I expect him to throw the ball well," Young said. "He was probably easing himself back with his velocity, but he located the ball well, commanded the ball well and mixed in his offspeed pitches. So he still threw the ball well."
Most of the Rangers offensive work came in the fourth inning, when they were trailing, 5-0. Arias reached on an infield single to lead off the inning, but Young grounded sharply into a double play. Josh Hamilton and Milton Bradley followed with a pair of singles before Hank Blalock struck out to end the inning.
The Rangers finished with just six hits and they are 4-24 when they get just six hits or fewer in a game this season.