David Lee had 19 points and 16 rebounds, and Danilo Gallinari had his strongest outing of a poor preseason with 18 points for New York (4-2).
Rajon Rondo scored 20 points for the Celtics (5-2), who had reserves on the court down the stretch of a game that was close throughout. One of them, Marquis Daniels, finished with 17 points.
Paul Pierce had 14 points, while Kevin Garnett and Ray Allen each were limited to 11 as Celtics coach Doc Rivers used his three stars very little in the second half with a week to go until the regular season starts.
Daniels tied it for the final time with two free throws before Harrington nailed his 3-pointer. Robinson followed with a layup after a turnover to make it 108-103 with 1:18 to go, and the Celtics' backups kept misfiring from there.
Robinson, back in a reserve role after a couple of preseason starts, had 16 points in 16 first-half minutes.
New York had struggled offensively throughout the preseason, shooting 38 percent and averaging 96 points while too often settling for jumpers in its first five games. The Knicks still hit only 44 percent in this one, but were able to play at the pace they want to much more than in previous games.
Just like their 96-82 loss in Boston on Oct. 9, the Knicks shot miserably at the start, missing 11 of their first 12 attempts. They rallied to tie it at 26 after one quarter, but the Celtics went back ahead in the second and took a 62-59 halftime lead behind 14 points from Rondo.
Up two after the third quarter, the Knicks scored nine straight points early in the fourth to open a 98-90 lead on Wilson Chandler's jumper. The Celtics answered with an 11-1 surge, with Daniels scoring the final four points for a 101-99 lead with 3:36 remaining, and it stayed close from there.
The Knicks close their exhibition schedule Wednesday night at New Jersey. The Celtics face the Cavaliers on Wednesday in Columbus, Ohio, then return to Ohio next Tuesday to visit the Cavs for the NBA season opener.