Byrd, who had already doubled and homered to help Texas to a one-run lead, was too far ahead of a changeup and grounded into a double play to end the Rangers' fifth inning.
"That was a big momentum shift," Angels manager Mike Scioscia said.
The first five Los Angeles hitters then reached to start the sixth, including back-to-back homers by Juan Rivera and Kendry Morales, and the AL West-leading Angles won their sixth straight game, 5-2 over the struggling Rangers on Monday night.
O'Sullivan (2-0) allowed consecutive homers to David Murphy and Byrd in the third. The Rangers were still up 2-1 when Byrd came up again with two runners on.
Byrd hit a grounder to third baseman Chone Figgins, who started the inning-ending double play.
"I believe it changed the game. ... I've got to get those runs in, at least one," Byrd said. "They came back out and do what they do. They just capitalize on mistakes and they don't make many mistakes."
The Rangers have lost eight of 11, a slide that cost them the division lead they had held since May 5 -- by 5½ games four weeks ago. Los Angeles, which has won 13 of 16 overall, took over first place Saturday and now leads by 2½ games.
Brian Fuentes worked the ninth for his major league-best 22nd save in 25 chances.
After being swept in a three-game series at Texas in mid-May, the Angels were 4½ games back. But that was when they were still without Guerrero, and the Rangers were still hitting.
"Over the last month, we've been playing better defense, our pitchers have been getting the job done and timely hitting," Torii Hunter said. "We're on all cylinders right now."
Texas is 10-15 in June and has scored two runs or less in 10 of those games. The Rangers had one hit in a 2-0 loss to San Diego on Sunday night, the second time they have been one-hit this season.
The Rangers doubled that total in the first against the Angels with hits from Michael Young and Byrd, but didn't capitalize. Nelson Cruz was intentionally walked to load the bases with two outs and Julio Borbon struck out in his first major league at-bat hours after being called up.
Murphy and Byrd hit 400-foot homers with two outs in the third. After Murphy homered to straightaway center, Byrd hobbled out of the batter's box when he fouled a pitch off his left knee. Byrd got a visit from manager Ron Washington and a team trainer before stepping back in and hitting the next pitch into the left-center seats.
"I didn't feel like I was doing my job to keep our team in it," O'Sullivan said. "I was a little frustrated with that, and really had to bear down."
O'Sullivan, in his third major league start since coming up when Ervin Santana went on the disabled list for the second time this season, struck out five and walked two. The right-hander has allowed six runs over 18 innings (3.00 ERA) and could remain in the rotation even when Santana, an All-Star last season, returns to the rotation.
"He's a young kid who pitched like a vet," Scioscia said.
Santana, out since June 12 with an inflamed right triceps, had no problems during a 45-pitch rehab assignment Sunday night in the Arizona Summer League. Scioscia said the right-hander will throw a bullpen session Wednesday and could return as early as Friday.
Padilla gave up five runs and 10 hits in five-plus innings, with five strikeouts and two walks. The right-hander had allowed only six runs over 19 innings in his previous three home starts.
Rangers CF Josh Hamilton, recovering from abdominal surgery, was 1 for 4 in a rehab assignment at Double-A Frisco. He had an RBI single, a stolen base and a run scored. ... Guerrero, who missed 35 games with a torn pectoral muscle, is closer to returning to the outfield. He is making longer throws from the outfield during pregame drills, and made some before Monday's game. He has been limited to DH duties since being activated May 25. ... The Rangers agreed to contract terms with 3B Tommy Mendonca from Fresno State, their second-round draft pick earlier this month. ... Morales has extra-base hits in 11 of his last 13 starts.