Maxey, the NBA's number two scorer this season, netted 21 points and Edwards came off the bench to score 22.
But it was Kelly Oubre's putback shot with 8.7 seconds remaining off the only miss of the night by Edwards that produced the winning points.
"Extremely tough. He fought in there," Maxey said of Oubre's final basket. "We got Justin the good look and KO got the rebound and put it back."
Boston's Derrick White missed a desperation half-court heave at the finish and the 76ers (7-4) improved to 2-1 over the Celtics this year, the games decided by a total of four points.
"That's how it goes," Maxey said. "We've played them 100 times it feels like since I've been in the NBA. They know what we're going to do and we know what they're going to do."
Maxey said winning tight games comes down to simple things.
"Resilient. We trust each other," he said. "Just try to make great plays and play great defense in the fourth quarter."
Edwards, a two-time state high school champion at Philadelphia's Imhotep High School, went 8-for-8 from the floor and 5-of-5 from three-point range in a sizzling display until his final miss that set up the winning basket.
Sixers reserve Quentin Grimes added 18 points while Boston's Jaylen Brown had a game-high 24 points.
Philadelphia was without Cameroonian big man Joel Embiid due to a sore right knee.
The Sixers led 23-22 after the lowest-scoring first quarter by each club this season and 51-41 at halftime but needed a Grimes three-pointer from half-court at the third-quarter buzzer to pull within 77-71, setting the stage for a fourth-quarter fightback.
Reigning NBA Most Valuable Player Shai Gilgeous-Alexander had 28 points and 11 assists to lead reigning champion Oklahoma City in a 126-102 home victory over the Golden State Warriors.
Chet Holmgren added 23 points and 11 rebounds for the Thunder, who boosted their NBA-best record to 11-1 with a third consecutive triumph.
Win streaks hit five
New York's Jalen Brunson scored 32 points, hitting six of nine three-point attempts, and passed off 10 assists to spark the host Knicks over Memphis 133-120.
The Knicks (7-3), who also got 21 points and 13 rebounds from Karl-Anthony Towns, stretched their win streak to five games and improved to 7-0 at home this season.
Also boosting their win streak to five games were the Denver Nuggets (8-2), who won 122-108 at Sacramento behind Serbian center Nikola Jokic's 35 points, 15 rebounds and seven assists.
The three-time NBA MVP averages 25.2 points, a league-best 11.9 assists and an NBA second-best 13.0 rebounds per contest.
Toronto's Brandon Ingram scored 25 points and Immanuel Quigley added 24 to lift the Raptors over host Brooklyn 119-106.
The Nets sank to 1-10, matching Washington with the NBA's worst record, and 0-6 at home.
Finland's Lauri Markkanen scored 35 points to lead the Utah Jazz over Indiana 152-128, the Pacers falling to 0-6 on the road and into the NBA basement at 1-10.