Harden scored 45 points, including a three-pointer to put the 76ers ahead with eight seconds remaining at TD Garden in Boston, BBC Sport reported.
“He was awesome,” said Sixers coach Doc Rivers. “Kept his cool, knew when to attack, knew when to give it up.”
Game two of the best-of-seven series is at the same venue on Thursday.
The Sixers were without the injured Joel Embiid. The six-time NBA All-Star suffered a knee injury in game three of the first-round play-off series against the Brooklyn Nets and it is not yet known when he will return.
Tyrese Maxey scored 26 points for the 76ers, while Jayson Tatum led the way for the Celtics with 39.
Two free throws from Tatum put Boston 115-114 ahead late in the final quarter, but Harden held his nerve to sink a three-pointer from 25 yards and give Philadelphia the lead.
“He had just the perfect mindset,” said Rivers. “I’m so happy for him because it just tells you what he can do on a given night.
“It’s amazing – the guy’s a Hall-of-Famer and all you hear is the other stuff about him. He’s fantastic.”
In the Western Conference, Nikola Jokic scored 39 points as the Denver Nuggets beat the Phoenix Suns 97-87 to take a 2-0 series lead.
“Nikola’s an MVP for a reason. He can take over a game. He can beat you in a lot of ways. I love an aggressive Nikola Jokic,” said Nuggets coach Michael Malone.
Aaron Gordon added 16 points and Kentavious Caldwell-Pope 14 for the Nuggets in a game dominated by defences at the Ball Arena in Denver.
Devin Booker scored 35 points and Kevin Durant 24 for the Suns, who lost Chris Paul to a groin injury in the third quarter.
Game three is in Phoenix on Saturday.