Making his first major league start since June 4, 2005, Ledezma (2-1) allowed six hits and two walks.
"This year, I think I've got more experience and more concentration," said Ledezma, who made 10 starts last season before being sent to Triple-A Toledo. "Always prepared for the first pitch, throwing a strike."
He used a fastball that reached 95 miles per hour and offspeed pitches to keep Cleveland off balance.
Detroit skipped Verlander (14-4) to give his tired arm a rest. He is scheduled to start Friday against the White Sox.
With their major league-leading 13th shutout of the season, the Tigers completed their 11th series sweep of the year. Detroit, which was outhit 8-6, improved its major league-best record to 75-36.
Craig Monroe hit a two-out RBI single in the fifth inning off Sabathia (8-8), who allowed an unearned run and six hits in seven innings, striking out six and walking two.
"We beat one of the best pitching performances I've seen against us all year," Detroit manager Jim Leyland said.
Brandon Inge led off the fifth with a bunt and reached when Sabathia couldn't handle it.
"I just tried to pick it up with my hand," said Sabathia, who was charged with an error. "I did the same thing in Boston (in his previous start)."
Inge advanced on Brent Clevlen's sacrifice and Placido Polanco's groundout, then scored on Monroe's single to left.
"Fastball down. I took a good swing," Monroe said.
Ledezma was replaced by Grilli after consecutive two-out singles by Jhonny Peralta and Shin-Soo Choo in the sixth. Following a wild pitch, Grilli retired Kelly Shoppach on a groundout.
Grilli walked No. 9 hitter Andy Marte with one out in the seventh, Grady Sizemore followed with a ground-rule double and Jason Michaels walked on a 3-2 pitch, loading the bases.
Walker relieved, struck out Travis Hafner and retired Victor Martinez on a forceout.
"We had the two guys up there we wanted with the bases loaded," Cleveland manager Eric Wedge said. "Walker really stepped up."
Magglio Ordonez tripled to lead off the sixth for Detroit. But with one out and runners at the corners, Vance Wilson failed on an attempted squeeze bunt and Ordonez was tagged out in a rundown.
- Cleveland 1B Casey Blake is day to day after spraining an ankle in Saturday night's game.
- Detroit's Ivan Rodriguez started at first base for the fourth time this season and in his career.
- Monroe was 2-for-4 and is batting .412 (21-for-51) in a 13-game hitting streak, the longest of his career.