Boston took its last lead, 81-80 with 4:24 left, when Travis Knight followed his own miss with a layup. Brevin Knight started a 9-0 Cleveland run with three free throws and Person hit two 3-pointers for an 89-81 lead with 1:31 remaining.
The Cavaliers led by at least four points the rest of the way.
Cleveland, which won its second consecutive game after losing its first two, was led by Shawn Kemp with 28 points and 15 rebounds. Derek Anderson added 17 points.
For Boston, winless in four games since beating Chicago in the season opener a week earlier, Antoine Walker had 22 points and 11 rebounds and Ron Mercer scored 16 points.
The game featured the two lowest-scoring, worst-shooting teams in the NBA but both made 42 percent of their shots. Cleveland began the game at 37.4 percent and 82 points per game, while Boston was at 38.5 percent and 83 points.
A 3-pointer by Travis Knight made it 93-88 and a layup by Dana Barros cut the Cleveland lead to 94-90. Brevin Knight ruined Boston's hopes with two free throws with 11 seconds to go. Mercer ended the scoring with a jumper with 3.8 seconds left.
The Celtics led 28-26 after one quarter but trailed 47-41 at halftime, the closest they've been at intermission in their five games this season.
Boston then scored the first six and last six points of the third quarter to lead 67-64. After Anderson's two free throws put Cleveland ahead 64-61 with 1:25 left, Boston scored on a free throw by Pervis Ellison, a layup by Walker, one free throw by Walter McCarty and two by Chauncey Billups. The Cavaliers had just three field goals in the quarter.
Notes: Eight of the 10 starters were first- or second-year players, including five rookies _ Cleveland's Anderson, Zydrunas Ilgauskas and Brevin Knight and Boston's Billups and Mercer. ... The game featured the coach of the 1996 NCAA champion Kentucky team, Boston's Rick Pitino, and four players from that squad: Anderson, Mercer, Walker and McCarty. ... Billups, scoreless in 11 minutes in a loss Wednesday night to Miami, started the game and had eight points in the first five minutes. ... Olympic gold medalist Carl Lewis attended the game. ... Cleveland's Bobby Sura missed the game with a sore ankle. ... Kemp got his 6,000th career rebound in the first quarter.