Norris (4-3) allowed two runs and six hits to snap a personal three-game losing streak and earn his first victory since Aug. 12 at Florida. He had been hit hard since becoming the first rookie in Astros history to win his first three starts.
"They let you know where you stand a little bit," Norris said of his last three outings. "You kind of gather yourself, get back to the basics, just keep going at it and that's exactly what I did."
The Astros, out of the NL playoff race, have won seven of their last eight home games.
"They keep playing all the time, they don't quit," manager Cecil Cooper said. "They play hard all the time."
Rollins drove Norris' fourth pitch into the Astros' bullpen for his fourth leadoff homer of the season and No. 32 for his career. Norris then struck out two of the next four hitters.
"The first batter of the game is always a question mark," Norris said. "But I came back and the rest of the game, I pitched really well."
Hamels extended his scoreless streak to a career-best 21 innings before the Astros got to him in the third. Bourn singled, stole second and scored on Berkman's one-out bloop single to right for the first run allowed by Hamels since Aug. 21.
The Phillies traded Bourn to the Astros in November 2007 as part of the Brad Lidge deal and are familiar with his abilities in the leadoff spot.
"With the type of speed he has, I think there's a reason he's been such a huge threat and a big player for their team," Hamels said.
The Astros scored three times in the fifth to take a 4-2 lead. Bourn beat out an infield single, stole second and advanced on Kaz Matsui's infield single. After Berkman grounded out, Lee blooped a two-run single to right. Tejada then drove in Lee with a double down the third-base line.
Once again, Bourn triggered the rally.
"He set the stage for the game," Phillies manager Charlie Manuel said. "He set up their offense a couple of times there. He can turn the game around. That's what speed will do for you."
Norris retired eight of the last nine hitters he faced. Cooper said Norris showed more poise than he had in his previous three starts.
"He's got the stuff, he just has to learn how to pitch," Cooper said. "Today, he kept his emotions out of it and just kept pitching."
The Astros called up Norris on July 29, when he was leading the Pacific Coast League in ERA and strikeouts. He pitched seven shutout innings in St. Louis, then beat Milwaukee and Florida before the losing streak.
Norris quickly found his rhythm against the Phillies and his confidence grew with every pitch.
"As the game progressed, I got into a groove," he said. "Once I got the ball rolling a little bit, I felt great. I want to keep it going for the next couple of starts."
LaTroy Hawkins allowed Werth's leadoff homer in the eighth before Valverde finished.
Bourn leads the NL with 52 stolen bases. ... The Phillies lead the NL with 94 home runs on the road. ... The Phillies dropped to 51-15 this season when Rollins scores a run. ... Cooper said INF Jeff Keppinger will miss a few days with tightness in his back.