Lester (2-2) allowed one run and seven hits for his second successive victory after opening the season with two losses. The left-hander struck out nine and walked none, winning an impressive pitchers' duel with White Sox ace Chris Sale.
"[Lester] was very good. Threw pitches for strikes, was down in the strike zone. He established an outstanding rhythm right from the start," Red Sox manager John Farrell said. "As well as Sale was pitching it needed everything that John gave us tonight."
The game was tied at 1 when Mike Napoli and pinch-hitter Mike Carp had back-to-back one-out singles in the ninth for the Red Sox. Ross then had a double to right against Ronald Belisario (1-2), driving home Napoli.
"Obviously when you've got a guy going like Sale on the other side and he's throwing the ball as well as he did tonight, it's good to grind out those innings and get some runs on the board late for us," Lester said.
Sale struck out 10 and threw a career-high 127 pitches in seven innings. The left-hander was trying to become the first White Sox pitcher to start 4-0 since Jose Contreras in 2006.
"Going out there and giving everything you got," Sale said about the number of pitches. "A team rolls in there like this you can't have any consternation with that, you can't shy away from that."
Sale and Lester provided a big lift for the tired bullpens on each team. The Red Sox used seven pitchers in a 6-4, 14-inning victory on Wednesday, while the White Sox used nine, including infielder Leury Garcia for one inning.
The White Sox had a chance to grab the lead in the seventh but Adam Dunn was thrown out at the plate when he tried to score from first on Alejandro De Aza's two-out double into the right-field corner.
"You've got to take a chance like that especially the way Lester's pitching," White Sox manager Robin Ventura said.
Sale got some help from center fielder Adam Eaton in the first inning. With two outs, David Ortiz drove a 2-2 pitch to deep left-center field, but Eaton jumped and took a home run away by making a one-handed grab.
That was the closest either team came to scoring until Bogaerts connected for his first home run of the season, driving a 1-0 pitch deep to left.
The White Sox responded in the bottom half. Flowers got on for Chicago's first baserunner and Garcia followed with a ground-rule double to right. Eaton then reached on an infield single, tying it at 1.
"If you're a baseball fan that was a really good game tonight," Ross said. "Two big lefties -- probably two of the best ones in the game going at it -- and we came out ahead."
Alexei Ramirez's two-out single in the ninth extended his season-opening hitting streak to 16 games, setting a Chicago record.
Napoli was back in the lineup after missing Wednesday's game with a dislocated left ring finger. ... Abreu, who was 1 for his last 21 entering the game, was replaced in the lineup by Paul Konerko. ... Red Sox OF Shane Victorino is set to begin a three-game rehab assignment Saturday at Triple-A Pawtucket and could join Boston for his season debut later next week. He was placed on the 15-day disabled list March 31 with a hamstring strain. ... Chicago purchased the contract of RHP Zach Putnam and designated LHP Donnie Veal for assignment. ... RHP John Lackey (2-1, 3.86 ERA) will start for the Red Sox on Friday in Boston against Baltimore RHP Chris Tillman (1-1, 0.84 ERA). ... RHP Felipe Paulino (0-1, 7.98 ERA) will pitch Friday for the White Sox in Texas against LHP Martin Perez (2-0, 2.70 ERA).