The A's went ahead to stay in their 9-3 victory Tuesday night on Norris' two-run double in the first off Perez, whose 26 consecutive scoreless innings over his previous three starts included a three-hit shutout in Oakland six days earlier.
"We just had to get the ball up in the zone. He's a great ground ball pitcher, he gets a lot of double plays," Norris said. "He was living down there in the zone, and we weren't biting on it and he had to elevate to get the strikes. ... When he got in a situation with runners on, we cashed in some runs."
When Norris added an RBI double to make it 4-0 in the third, the A's already had six hits off Perez (4-1), who started the night with an American League-best 1.42 ERA.
"Well, we got on him early today, and typically with a good pitcher like that who's on a roll, your best chance is early on before he gets settled in," Oakland manager Bob Melvin said. "If he gets into a rhythm as we saw last time, we're going to have a little trouble with him."
Scott Kazmir (4-0) needed 95 pitches to get through five innings but left with a 9-3 lead and tied for tops in the AL with his fourth victory. The lefty struck out four and walked one.
The eight runs allowed by Perez were two more than his combined total through his first five starts, and his ERA more than doubled to 2.95. He struck out three and walked three in 4 2/3 innings.
"They looked more aggressive tonight, and (a) couple of pitches (were) up," Perez said. "When I got the ball down, I got a couple of good outs. Tonight, it didn't happen how I wanted."
Oakland, swept in three games at home by the Rangers last week, has won the first two games in this series that started with the teams tied atop the AL West.
At 17-10, the A's have the best record in the American League. Texas (15-12) has its first three-game losing streak this season.
A's right-hander Sonny Gray, who lost in the matchup against Perez last week, reached four wins by throwing a three-hit shutout in the series opener. Gray beat Rangers ace Yu Darvish, who with 3 1/3 innings had his shortest start in three seasons since coming over from Japan.
Perez and Toronto's Mark Buehrle were the only other AL pitchers who went into Tuesday with four wins.
When Craig Gentry led off the second with a triple, the A's already had three hits -- matching through seven batters what they got against Perez last week. Gentry scored on a groundout by Josh Reddick, who added a two-run single in Oakland's five-run fifth.
"They were taking those low pitches, forcing him to bring the ball a little up in the strike zone," Rangers catcher Robinson Chirinos said. "They put a good swing on everything he was throwing up in the strike zone. I think that was the difference between last week and today."
Perez was gone after issuing a bases-loaded walk to Alberto Callaspo in the fifth. Pinch hitter Brandon Moss then drew a bases-loaded walk on an eight-pitch at-bat against reliever Jason Frasor before Reddick's single made it 9-1.
Alex Rios had an RBI double for Texas in the fourth but was thrown out trying to come home on a grounder. After the Rangers had consecutive singles to start the fifth, Elvis Andrus sent home a run on a groundout and Prince Fielder had an RBI single.
"They made him work some," Melvin said of Kazmir. "I could have let him go out again, but they made him work pretty hard. This is the first game this year that it hasn't been real easy for him. But again, he ends up with the win, and if that's a bad game for him, we'll take it."
Norris is 10-for-17 (.588) during a six-game hitting streak. ... It was Texas manager Ron Washington's 62nd birthday. The Rangers are 0-6 on his birthday. ... Kazmir's ERA rose to 2.11 after he entered at 1.62, third-best in the AL. ... Oakland is 11-4 on the road, best in the AL. ... Perez's scoreless streak was the fifth-longest for a Rangers starter. ... There was delay of 1 minute, 38 seconds in the fifth for Washington's challenge after first-base umpire Laz Diaz ruled a foul ball on Andrus' liner down the right-field line just past the infield dirt. The call stood after replay, meaning there was no clear or convincing evidence to overturn it.