Tatum added 13 rebounds five assists and a couple of steals and the Celtics' suffocating defence neutralised newly minted NBA Most Valuable Player Joel Embiid as Boston booked a return date with the Miami Heat - the team they beat in seven games in last year's conference finals.
Jaylen Brown added 25 points for Boston, who broke open a close game in the third quarter as they out-scored the 76ers 33-10.
Philadelphia went six minutes of the period without a basket and trailed by 26 heading into the fourth, when Boston pushed the lead to 30 with less than four minutes to play.
His 51 points eclipsed the Game Seven record of 50 set by Golden State's Stephen Curry against the Sacramento Kings in the first round this month.
It was a bounce-back performance for Tatum, who struggled for long stretches in the series and endured a horrific shooting night until drilling four crucial three-pointers in the fourth quarter of the Celtics' series-extending Game Six win on Thursday.
He began fashioning his Game Seven masterpiece in the opening minutes.
The 76ers turned an early six-point deficit into a 29-23 lead through one quarter, while a nip-and-tuck second period saw the Celtics up 55-52 at halftime.
Then the Celtics blew it wide open in the third. Tatum scored 17 points in the period, draining four of his six three-pointers as the shell-shocked 76ers struggled, and failed, to find a response.
"Going into Game Six I was too locked-in, I was too tight, thinking about what I need to do, how many points I need to score," Tatum said.
After a lighthearted practice on Saturday, Tatum said he was more himself.
"That's when I play my best is when I'm having fun."
It was another crushing playoff exit for the 76ers and Embiid, who was held to 15 points on five-of-18 shooting.
Philadelphia's Doc Rivers fell to 6-10 in Game 7s as a coach. That's the most Game 7 defeats of any NBA coach in history and Rivers remained without a Game Seven win on the road in his coaching career.
The 76ers had the third-best regular-season record behind the Milwaukee Bucks and Boston, but even with Embiid enjoying an MVP season, it failed to translate into post-season success.
They have reached the playoffs for six straight seasons but haven't made it out of the second round.
Tobias Harris led the 76ers with 19 points but Boston limited the Sixers to 37.3% shooting from the floor and harried them into 14 turnovers.
Turnover trouble
Star guard James Harden scored just nine, making three of his 11 shots from the field.
Boston veteran Al Horford did a terrific job guarding Embiid. He scored just six points but grabbed 10 rebounds with two steals and three blocked shots.
Rivers said the rot set in for the 76ers in the second quarter.
"We were playing so well. We trusted, the ball was moving, and then we had I think it was three or four turnovers in a row," Rivers said.
Rivers pointed to a flagrant foul called on Harden after his arm hit Brown in the face as the ball slipped out of Harden's hands on a layup attempt.
Instead of going up by 10, the Sixers gave up the ball.
"After that, we never played right again," Rivers said.
The Celtics will host the Heat in Game One of the Eastern Conference finals on Wednesday with an NBA Finals berth on the line.
Last year the Heat were the top seeds in the East. This year they are the first eighth-seeded team to reach the conference finals since the 1999 Knicks.
