"That's something that we needed," Byrd said.
The Rangers were leading by one with two on and two out in the fourth when Byrd hit his second homer and seventh of the season.
With Texas up by one Monday, in part because of his homer, Byrd hit into an inning-ending double play with two on. He was 3-for-4, but blamed himself for the 5-2 loss because right after his grounder, the next five Angels reached, including back-to-back homers.
Byrd made up for that, though he insisted he wasn't thinking about it Tuesday.
"I go home, it's all gone," Byrd said. "I have the drive home to be mad. Once I get home, that's done. I move on and try to do what I did."
Texas quickly built a 3-0 lead, scoring more runs through its first three batters off Saunders (8-5) than in 10 of its June games. Kinsler extended his team record with his 10th career leadoff homer, fifth this year, and Byrd connected after Michael Young walked.
Saunders is 0-5 with an 11.67 ERA in five career starts at Rangers Ballpark. The left-hander is 21-4 with a 2.92 ERA in 31 starts as a visitor to other ballparks -- and 3-0 with a 1.44 ERA in four home starts against Texas.
"Maybe you guys can figure it out because I haven't," Saunders said. "It seems like every time I come and pitch here, I don't have any command or they're sitting one way and I'm pitching right into their swings."
Nelson Cruz and David Murphy also went deep for Texas, which ended what had been one of the worst offensive months in team history with a season-high five home runs -- four in 3 2/3 innings off Saunders. Elvis Andrus had four stolen bases, the first AL rookie since 1990 to do that.
"Obviously he doesn't feel comfortable. I don't know if it's the mound, a good offensive club," manager Mike Scioscia said. "He has trouble executing pitches as consistently as he can."
The Angels had their six-game winning streak snapped, but are still 1 1/2 games ahead of Texas.
Scott Feldman (6-2) made it through six innings even after throwing 57 pitches the first two innings while giving up a three-run lead. The right-hander retired 13 of the last 14 batters he faced, 11 in a row after Torii Hunter's RBI single tied it at 3 in the second.
"I wasn't sure how far he was going to be able to go after that," manager Ron Washington said. "He settled down and started pitching."
And the Rangers hitters were slugging again.
Cruz snapped a 2-for-27 slide with a go-ahead homer in the third. Murphy's fourth homer in six games came in the seventh and made it 9-3.
"Hopefully, this will create some momentum for us," Murphy said.
The Rangers entered the final game of June hitting a majors-low .221 for the month. They also blew the division lead that was once as high as 5 1/2 games by losing eight of 11 while the Angels won 13 of 16.
Saunders gave up four homers for the first time in his 80 career starts (40-20). He gave up three long balls and seven runs in 5 1/3 innings at Rangers Ballpark on May 15.
The Angels had five singles in the second to score three times. Erick Aybar was called out for stepping out of the batter's box on a bunt, and Feldman got out of the inning when Vladimir Guerrero popped out to short.
"When I got in the dugout, [my goal] was get through six," Feldman said. "I had to work quick and conserve my pitches."
The last AL rookie with at least four steals in a game was Baltimore's Luis Matos on July 30, 2000, when he had five against Cleveland. Colorado rookie Dexter Fowler had five steals against San Diego earlier this season. ... Andrus stole second and third in the fifth. He is the fourth Texas player with at least four steals in a game. The team record is five by Scarborough Green on Sept. 28, 2000.