Boston had looked to be cruising towards their seventh straight victory after dominating Milwaukee for long periods at the TD Garden.
Exploiting the absence of injured Bucks talisman Giannis Antetokounmpo, the Celtics surged into a 21-point lead late in the third quarter after Milwaukee had closed to within five just after half-time.
But a blistering fourth-quarter performance by Milwaukee - who outscored the Celtics 36-21 in the final frame - set up a nerve-shredding finale before Boston squeaked home.
"It was ugly," Tatum admitted. "We didn't play the way we wanted to in the fourth quarter, and they went on a run.
"No lead is safe in the NBA. But a win is a win and we'll take it."
Boston improved to 55-14 with the victory, leaving them a hefty 11 games clear of the second-placed Bucks.
The Celtics now need just three more wins to be assured of top-seeding and home-court advantage through the play-offs.
Even allowing for the fourth-quarter wobble, few would bet against them returning to the NBA Finals in June after a performance which once again showcased their balanced offence.
'Long way to go'
Five Boston players finished with double-digit points tallies, including Derrick White (23), Jaylen Brown (21) and Kristaps Porzingis (17). Payton Pritchard added 19 off the bench.
"We've just got a really complete team," Tatum told ESPN.
"We've been playing great all season, but we've still got a long way to go," Tatum said.
"I like the way that we're trending, but we're just focused on getting better every single day.
"We're not looking to the play-offs. We've got a game on Friday against Detroit - and that's the mindset that we've got to have."
With Antetokounmpo absent, Milwaukee's challenge was spearheaded by Damian Lillard, who finished with 32 points while Bobby Portis added 24 off the bench.
In other games on Wednesday, Kevin Durant surpassed Shaquille O'Neal to move into eighth place on the NBA's all-time scoring list as the Phoenix Suns eased past the Philadelphia 76ers 115-102.
Durant, eight points behind O'Neal's tally of 28,596 points before the game, drained a jump shot in the third quarter to move past the former Los Angeles Lakers icon.
"Well deserved," O'Neal said in a video message to Durant. "It doesn't stop here ... If anybody was gonna pass me, it was definitely gonna be you."
Durant finished the night with 22 points, while Grayson Allen led the Suns scoring with 32 points including nine three-pointers.
In San Francisco, the Golden State Warriors bounced back from Monday's defeat to the New York Knicks to hand the Memphis Grizzlies a 137-116 drubbing.
The Warriors improved to 36-32 and remain 10th in the West.
Butler reminder
In Cleveland, the Miami Heat produced a barnstorming finale to pip the Cavaliers 107-104.
Miami led for most of the game but Cleveland crept into a four-point lead with one minute and 34 seconds remaining in the fourth.
The Heat's Terry Rozier rattled off nine points in the closing stages including a crucial 26-foot three-pointer with 15 seconds remaining to restore Miami's advantage and see the visitors home.
Rozier finished with 24 points while Jimmy Butler led Miami's scorers with 30.
"That's the most important thing, just getting the win," Rozier said.
"My teammates and coaches do a great job of reminding me daily, night in, night out, who I am and just to be that. Happy we got the win."
Miami improved to 38-31 with the victory and sit in seventh place in the East. Cleveland remain third in the table at 43-26.
The Sacramento Kings eased past the Toronto Raptors 123-89, while the Indiana Pacers swatted aside Detroit 122-103.
Western Conference leaders Oklahoma City breezed past the Utah Jazz 119-107, with Chet Holmgren (35) and Shai Gilgeous-Alexander (31) combining for 66 points.
