After Garnett missed a baseline jumper, Irving dribbled past midcourt and drove to the basket for a three-point lead with 52 seconds remaining.
Following another miss by the Celtics, the Cavaliers called time out. Irving dribbled into the lane and hit a twisting shot while being fouled by Jared Sullinger. He made the free throw to put Cleveland ahead 93-87 with 22.8 seconds left.
After Pierce shot an airball, Irving made two more free throws with 18.2 seconds left to put the game away.
Tristan Thompson added 21 points for the Cavaliers.
The Cavaliers acquired three players and a future first-round draft pick from Memphis earlier in the day. Cleveland picked up forward Marreese Speights and guards Wayne Ellington and Josh Selby in exchange for forward Jon Leuer. Speights will help replace Anderson Varejao, who will miss the remainder of the season after developing a blood clot in his right lung. Varejao had had surgery on Jan. 10 to repair a split leg muscle. He was expected to miss two months, but the blood clot was discovered last week and he has been placed on blood thinners.
Irving, who was 7 of 20 from the field in Saturday's loss to Utah, had a strong start against Boston with 19 points in the first quarter, hitting 8 of 10 from the field.
Cavaliers starting guard C.J. Miles didn't play because of an illness that has hit several players and a couple of assistant coaches.
Sullinger, who played at Ohio State, scored 12 points and had 10 rebounds for Boston.
The Cavaliers' new players were expected to arrive in Cleveland on Tuesday night. They must first pass physicals and Scott hopes they will be able to practice Thursday and possibly play Friday against Milwaukee.
NOTES: Bradley, who scored seven points, continues to play with sore ribs. "I feel for Avery because that's a hard injury to play through," Rivers said. "It hurts to breathe and it hurts every time you get hit anywhere near your ribs." Scott has traded text messages with Varejao, who remains hospitalized in the Cleveland Clinic. ... The Cavaliers waived G Jeremy Pargo and recalled C Kevin Jones from Canton of the Developmental League.