Coincidentally, Eaton is 3-0 in games that have followed those embarrassing losses.
"That's baseball. You can bounce back and start all over from scratch the next day," manager Charlie Manuel said.
"Sometimes we play 20 times in a row," Eaton said, "so there's going to be ups and downs."
The Phillies, who denied Ozzie Guillen his 300th win as the White Sox' manager, will try to go three games over the .500 mark for the first time this season when Cole Hamels (8-2) faces Sox righthander Jose Contreras (4-6) on Tuesday.
In his second straight sharp outing, Eaton allowed four hits and no runs in seven innings.
"He started hitters more with change-ups and sliders," Manuel said. "He had problems putting his fastball where he wanted it, but he battled through it."
Eaton, who made 100 pitches and struck out five, said he "tried to stay aggressive and force them to put the ball in play early [in the count]. I didn't care how many strikeouts I had."
Besides his superb pitching, Eaton just missed his third career homer, but his third-inning drive was caught in front of the left-field wall.
"It's a little bit of a Little League field here," Eaton said. "I hit it off the end of the bat but still got it to the track."
Burrell gave Eaton a 1-0 lead when he sent Javier Vazquez's 2-1 slider into the seats in left-center with two outs in the second. It was his eighth homer of the season and the 196th of his career, putting him into sole possession of seventh place on the Phillies' all-time list. Burrell had been tied with Bobby Abreu.
"He's always pitched well against us and it seems like he has the velocity back that he had in Montreal," Burrell said of Vazquez, "so it was important to get the lead."
Two innings after Burrell's homer, Howard swatted a majestic drive into the second deck in right, putting the Phils ahead, 2-0. After a slow start that included a stint on the disabled list, Howard has 13 homers and is on a pace to hit 33.
Eaton escaped a first-and-second, no-out jam in the fifth, helping his own cause by racing toward home and making a diving catch of Vazquez's one-out bunt.
In the next inning, Eaton struck out Jermaine Dye (swinging at a 92 m.p.h. fastball) and retired Juan Uribe on a left-field fly to squash a first-and-second, one-out threat as the White demonstrated why they are last in the majors in batting average (.232).
Rollins gave the Phils a 3-0 sixth-inning lead by slamming a one-out drive to right for his 11th homer.
In six innings, Vazquez allowed just four hits, but three of them went for homers. Vazquez entered the game having served eight homers in 72 innings.
The Phils, winners of seven of their last 10, threatened in the seventh, but Wes Helms, pinch-hitting for Eaton, bounced into an inning-ending double-play with runners at first and third.
By lifting Eaton while he had a shutout, Manuel had a chance to look like, well, a Meathead - as in the nickname Archie Bunker used for the All in the Family character played by Rob Reiner, who threw out tonight's ceremonial first pitch.
But solid relief work by Mike Zagurski and Antonio Alfonseca gave the Phils their second shutout in the last three games and prevented any second-guessing.
Contact staff writer Sam Carchidi at 215-854-5181 or firstname.lastname@example.org.