Smart is expected to be sidelined six to eight weeks, the team said.
There is hope that Smart can return in six weeks, sources told ESPN's Adrian Wojnarowski, which would mean he'd be available for the Eastern Conference playoffs.
Smart suffered what the team initially termed a sprained right thumb while diving to the floor during the first half of a loss to the Indiana Pacers on Sunday. He had the thumb taped and finished the game, after which he downplayed the severity of the injury.
Initial tests Monday showed more worrisome damage to the thumb, forcing Smart to examine his options. He traveled to New York on Thursday to seek a second opinion before electing for surgery that might allow for a playoff return.
Smart, a restricted free agent this summer, is averaging 10.2 points, 4.8 assists and 3.5 rebounds over 29.9 minutes per game in 54 appearances this season.
He missed 11 games after punching a glass picture frame in Los Angeles in January, but he had given the Celtics a much-needed boost since his return after the All-Star break.
Smart had said that missing time with the original hand injury had made him more appreciative of being on the court.
"I love this game. I love to play," Smart said. "So when that kinda gets taken away from you, temporarily or permanently, you kinda start realizing how much you do miss this game and how much it is easy to take it for granted when you can't do that."