Pedroia, the reigning American League MVP, went 1-for-3 with two walks and a stolen base and drove in Drew with the game's first run.
The top-of-the-lineup fireworks complemented the effort on the mound by southpaw Jon Lester (7-6). Lester improved to 8-0 against Baltimore (34-42) with seven shutout innings, scattering five hits and striking out eight without walking a batter.
"Maybe their lineup just fits me for some reason, I don't know," Lester said. "I don't pitch any different against them than against any other team, but I'm just fortunate to get some runs early on and that's the biggest thing."
Lester came into the game with a 4.68 ERA, but he looked like an ace from the outset.
"When things were going a little rocky, he and [pitching coach] John Farrell went out there and would put their heads together and keep working and boy, we're seeing good results now," said Francona, who was especially impressed with Lester's breaking pitch on Monday. "The depth and the tight spin, it's almost an attack pitch now, not a 'get it over for a strike' pitch."
Relievers Manny Delcarmen, Hideki Okajima, Ramon Ramirez and closer Jonathan Papelbon preserved the shutout, with Papelbon retiring Orioles rookie Matt Wieters for the final out to notch his 132nd career save, tying the Red Sox record held by Bob Stanley.
That milestone was set in exciting fashion. Papelbon came on for Ramirez with two outs and two on to face Wieters, who hit a shallow fly ball to left which was caught by a diving Jason Bay.
"I was already thinking about what my first pitch would be to [on-deck hitter] Luke Scott," Papelbon said. "It's an exciting feeling, but I'm going to try to keep focusing on what I have to do, stay consistent with my work."
Drew, who found out he would be leading off at the end of batting practice, opened the game with a triple off the center-field wall, and gave the Red Sox a quick 1-0 lead as he scored on a single up the middle by Pedroia.
Though the Red Sox threatened in each of the next two innings, getting runners as far as third base, they didn't score again until the fourth. Jason Varitek's RBI single to center scored Jacoby Ellsbury, who'd been hit with a pitch to lead off the inning.
Drew, though, delivered the big blow, a two-run homer to center, his 10th of the season, to make it 4-0.
Drew would single in his fourth at-bat, coming to the plate for the last time in the eighth against Orioles reliever Chris Ray. That at-bat resulted in an infield groundout, but Drew joked afterward that he had definitely had double on the mind.
"Absolutely, there's not a player alive that wouldn't know that situation," Drew said. "I tried to hit a double, it just didn't work out. I was just going to hit the ball and run straight to second, right through the middle of the infield, if I had to."
Pedroia, who hit .326 with 17 homers and 54 doubles en route to AL MVP honors last year, moved into the leadoff spot on May 31. But since the move, his average had dropped 40 points, from .328 at the time of the original shift to .288 heading into Monday's game, and Francona decided to make the move.
"I just think he was trying too hard," Francona said. "We just want him to be himself, and I think if we move him back to the two-hole, that will come naturally."
Pedroia, who was 8-for-26 during the first seven games of the 10-game road trip and 3-for-12 in the most recent three-game Atlanta series, responded to the shift.
"I think I've been trying to do too much, and it backfires, especially when you have a lot of energy," said Pedroia. "When you hit leadoff, you want to get on base so bad. And I'm a little bit of a hacker."
Pedroia had no problem with the move, and he gave credit to Francona for the idea.
"He said, 'We'll flip-flop you and J.D. and just go do your thing,'" Pedroia said. "It seems like every time he mixes things up, we do something good for him."
On his end, though, Francona deferred the credit.
"You know what I'm going to take credit for? Just having good players," Francona said. "That got some reaction in the dugout in the first inning, as you can imagine."
Even before the game, though, Boston's skipper had downplayed the significance of the move.
"It's not the biggest deal in the world," Francona said. "This wasn't something I spent all night thinking about, I just thought it made sense."