Boston also got a boost from Jae Crowder, who posted a career-best 22 points on 9-of-14 shooting to go along with four steals and a pair of blocks. His swat on Davis with 29 seconds remaining helped the Celtics open their three- game homestand with a win.
Davis filled the stat sheet with 34 points, nine rebounds, four assists and three blocks in New Orleans' fifth straight loss against the Eastern Conference.
Tyler Zeller opened the fourth quarter with a reverse layup and Marcus Thornton followed with a four-point possession -- a 3-pointer and technical free throw -- to give the Celtics their first lead since early in the third at 78-76.
They led 101-100 with less than a minute to play when rookie Marcus Smart hit the biggest shot of the game. Avery Bradley drove and pulled up for a short jumper, but had his shot turned aside by Davis. Bradley tracked down the rebound and kicked to Smart, who drained a corner 3 to make it a two- possession game.
Davis was blocked by Crowder at the other end, and Sullinger hit four free throws to seal the outcome.
Both teams held narrow leads during a see-saw first half, which ended in a 49-49 draw. It was 22-22 through 12 minutes.
The Pelicans shot 50 percent in the third and created some distance. Their lead reached six points on two occasions before taking a 76-72 lead into the fourth.
Davis shot 14-of-15 from the free throw line ... The Pelicans opened a five- game road trip against Eastern Conference teams... Boston had lost five straight to New Orleans ... Earlier Monday, Boston and New Orleans were involved in a five-player, three-team trade that sent forward Jeff Green to the Grizzlies. The Celtics acquired forward Tayshaun Prince from Memphis and guard Austin Rivers from New Orleans, while the Pelicans added Quincy Pondexter from the Grizzlies and Memphis acquired guard Russ Smith from New Orleans. Draft picks were also exchanged.