McCarthy (5-2) was coming off his first major league shutout on May 24 against Houston, but he walked the game's first two batters before Matt Holliday grounded into a double play on a 2-0 pitch.
"We had our chances to bust it open," A's manager Bob Geren said.
McCarthy settled down to retire 12 in a row before giving up a fifth-inning single. He allowed one run and three hits in six innings with two strikeouts and three walks as he matched his career high for victories.
"I was all over the place in the first," said McCarthy, in his second season as a full-time starter. "I couldn't find what I wanted to do. I was a little out of sorts. But we figured it out. In the second I had better command and went from there. A couple years ago, I don't know if I get out of that jam."
By working out of trouble, McCarthy earned the confidence of his teammates.
"That was big," said Michael Young. "He settled back down. That's what pitchers do when they get experience, make the right adjustment. It's a credit to him that he stayed with it and pitched a real good game."
Texas went in front with a three-run third on Young's run-scoring double and Josh Hamilton's two-run single.
Blalock and Byrd led off the fourth against rookie starter Brett Anderson (2-5) with homers and Young added an RBI double later in the inning to stretch the lead to 6-1.
Nelson Cruz's 445-foot solo blast in a four-run fifth was his seventh homer in his last 11 games, helping the Rangers extend their edge to 10-1.
Ian Kinsler had four RBIs, including a two-run double highlighting a three-run seventh to make it 13-1.
Byrd, Young and rookie Elvis Andrus each had three hits for Texas.
The Rangers have been winning with pitching and defense, but on Saturday night they got a lift from their 17-hit attack.
"Our approach has been a little off, so we've made it a point to have better at bats," Young said.
Texas (30-19) has won seven of nine to equal the 1996 team for the second-best start in club history. In 1998, Texas opened at 31-18 for its best start.
The Rangers are 20-8 in May, setting a club record for wins in that month.
The injury-depleted A's (18-29) lost their fourth in a row to match the fourth-worst start (1997) in the team's 42-year run in Oakland.
Anderson took a step back against Texas, giving up six runs and eight hits in four innings.
"The crispness on his pitches were down, his energy was down," Geren said. "He had a bit of a cold but that wasn't the main reason. He just didn't have his best stuff. I fully expect him to bounce back."
Texas is 13-2 against the AL West this season. ... Rangers manager Ron Washington started Blalock at 1B instead of Chris Davis after Davis went 0-for-7 with six strikeouts in Friday's doubleheader. Davis, who leads the majors with 77 strikeouts, arrived at the ballpark six hours before first pitch to work with hitting coach Rudy Jaramillo on swing mechanics. ... The Rangers will start Vicente Padilla, Scott Feldman and McCarthy in the three-game road series against the Yankees that opens on Tuesday night. ... The Rangers have had eight straight home crowds of 30,000 or more, the longest streak since nine in a row in 2005.