Carmona pitched a five-hitter for Cleveland's second consecutive shutout over the struggling Toronto Blue Jays, a 3-0 victory in Monday's doubleheader opener.
Lee, 6-0 with a major league-leading 0.81 ERA, started the second game and tried to complete a sweep for the Indians.
"Fausto was fantastic," Indians manager Eric Wedge said. "He got better as the game went on -- noticeably better."
Carmona (4-1) struck out three and walked four in his second shutout and third complete game in 47 big league starts. The right-hander was 9 months old the previous time Cleveland pitched consecutive shutouts at home, on Sept. 8-9, 1982, against Boston and Baltimore.
It had been 36 years since the Indians did it in Cleveland against the same opponent, blanking the Chicago White Sox on May 6-7, 1972.
"We know our starting pitching is extremely strong," said David Dellucci, whose two-out single in the sixth drove in the first run. "The way Fausto was pitching, we knew it wouldn't take a lot."
A.J. Burnett (3-4) gave up three runs and five hits in 7 2/3 innings for the Blue Jays, who have been blanked in 22 consecutive innings.
"It's tough, but that's the way it has been going," Toronto manager John Gibbons said. "Carmona was great. He can do that to you."
Cleveland won for the fifth time in six games and extended the Blue Jays' losing streak to four. The Indians outscored Toronto 21-1 in the first three games of the series, and the Blue Jays have lost nine straight games in Cleveland.
Shaun Marcum started for Toronto in the second game. Sunday's rainout caused the first single-admission doubleheader at Cleveland since Aug. 13, 1997.
Burnett allowed only one hit until Grady Sizemore doubled off the wall in right-center with one out in the sixth. Sizemore went to third on a groundout by Jamey Carroll and scored on Dellucci's single.
"That's a big two-out knock in a zero-zero ball game," Wedge said.
Cleveland's defense helped Carmona. In the second, Cabrera ranged behind second base to field a sharp grounder, shoveled the ball with his glove to second baseman Jamey Carroll, who turned the double play.
Burnett dropped to 30-34 with a 4.31 ERA when starting with more than four days' rest. He's 42-36 with a 3.43 ERA when starting on normal rest. ... Stairs had three of Toronto's hits. ... Carmona fanned Aaron Hill three times -- Carmona's only strikeouts. ... Burnett is 1-4 in five starts against the Indians, including 0-4 in Cleveland. ... Indians OF Shin-Soo Choo, recovering from elbow surgery last September, is expected to begin a rehab assignment this week at Triple-A Buffalo.