"It's a tremendous relief," Cubs manager Dusty Baker said.
"Like I always say, 130, 140 pitches for me is nothing," Zambrano said. "I work hard and that's why I'm here."
Zambrano (3-1) put men on in every inning but the seventh and eighth but struck out five and was helped by double plays in the third and fourth to win for the first time since April 20 at St. Louis.
Bobby Abreu walked to start the ninth and reached third with one out. Zambrano then got Ryan Howard to ground to first before getting Offerman on a foul tip. Zambrano was still throwing 96 mph in the ninth.
"It was his [game]," Baker said. "I know he threw a lot of pitches but he's strong as an ox. We're going to have to watch him the next start, maybe give him an extra day or something. He wanted it."
Ryan Dempster was warming up in the ninth, a day after he was told he was becoming the new closer. Dempster, 1-3 as a starter, was scheduled to start Monday against the Mets, but the Cubs told him of the switch Saturday. Dempster pitched in the bullpen late last season after returning from elbow surgery.
"It's exciting," Dempster said. "I don't know what to say. Just go out there and when they call on me to do the job, try not to let anyone score."
Baker has said the struggles of his closers have left him sleepless at night and said this move is definite until Joe Borowski returns from a rehab stint in Triple-A Iowa.
"Then we'll make a decision," he said.
Brett Myers (2-2) struck out 10 and also gave up just five hits -- but two of them were homers in the fourth. The Phillies were trying to sweep the three-game series.
"One day I'll probably give up six runs and get the win," Myers said. "That's baseball. Right now we're not scoring as many runs as we'd like."
Perez led off the fourth with a drive to right, his fourth, and, after Derrek Lee struck out, Ramirez hit his sixth to make it 2-0.
"We needed it," Ramirez said. "Seven games in a row. It's big for us, we've got to keep it going."
Myers was truly hit-or-miss from the third through fifth innings. He struck out eight of 11 batters, fanning the side in the fourth and fifth.
"He did a hell of a job," Phillies manager Charlie Manuel said. "I actually thought he was pretty strong at the end, too."
Myers was hit in the side of his head by Michael Barrett's single in the second inning.
"It stung, but you've got so much adrenaline rolling through you, you feel it but I probably won't really feel it until tonight," Myers said. "It got me flush. Right now, I'm OK."
Abreu's two-out homer in the sixth brought the Phillies within 2-1.
Ramirez led off the bottom of the seventh with a single, but after Jeromy Burnitz flied out, Ramirez was caught trying to steal second. Barrett then doubled to left past a diving Abreu in right and Myers intentionally walked Ben Grieve before striking out Jerry Hairston.
Lee went 0-for-3, ending his 18-game hitting streak.
Right-hander Jon Leicester is getting recalled from Triple-A Iowa to start in Dempster's place Monday ... The Cubs signed left-hander Darren Oliver to a minor-league deal on Friday and he started at Iowa on
Sunday, giving up nine runs and nine hits in 2 1/3 innings. ... After a number of Cubs shaved their heads in the past few days, Baker shaved his goatee at his wife's behest to change the team's luck. "I usually don't listen when she tells me stuff, but I listened this time," he said. ... Greg Maddux threw the previous complete game for the Cubs on July 22, 2004, against Cincinnati.