Tatum hit 14-of-23 from the floor, 5-of-10 from three-point range, and grabbed nine rebounds in more than 47 minutes while playing tough defense against Kawhi Leonard to send the Celtics into the NBA All-Star Game break as winners.
"This was a tough one," Tatum said. "It's nice to go to the All-Star break with a win and enjoy those moments off."
Tatum's hot night came after a 5-for-15 shooting effort and 0-for-7 three-point performance earlier in the week against Houston.
"Just a lot of hard work," Tatum said of his sizzling shooting. "My teammates trust me and vice versa. We do a good job of getting the hot guy the ball whoever it is."
Tatum struck for two late hoops in overtime, a layup over Leonard with 24.8 seconds remaining to put Boston ahead 127-124.
But Landry Shamet's three-pointer pulled the Clippers level and sent the game into a second extra period, which the Celtics dominated to claim the triumph.
In the only other game on the last night of contests before the NBA All-Star Game break, Italian forward Danilo Gallinari scored 29 points on 12-of-18 shooting to lead the Oklahoma City Thunder to a 123-118 victory at New Orleans.
German reserve Dennis Schroder added 22, New Zealand center Steven Adams contributed 11 points -- including his first NBA three-pointer -- and 11 rebounds.
Chris Paul had 14 points and 12 assists for the Thunder.
Pelicans rookie star Zion Williamson scored a game-high 32 points in his 10th NBA contest while JJ Redick added 24 points off the bench for New Orleans.
- Celtics honor Garnett -
The Celtics announced Thursday that Kevin Garnett, who helped Boston capture the 2008 NBA crown, will have his number 5 jersey retired next season, the 24th former player so honored by the club.
"I'm honored and thankful to have my number retired with the Celtics," Garnett said.
He was the 2008 NBA Defensive Player of the Year and an NBA All-Star in five of his six seasons with the Celtics.
In 396 games for Boston, Garnett averaged 15.7 points, 8.3 rebounds, 2.7 assists, 1.2 steals and 1.0 blocks a game.