Harris was one of four Detroit players to finish in double figures, with Jalen Duren adding 18 points and point guard Cade Cunningham scoring 16 points with 14 assists.
The win sees Detroit extend their lead at the top of the Eastern Conference to 31-10, 5.5 games ahead of second-placed Boston, who fell to 26-16 with the defeat.
Jaylen Brown led the Celtics in scoring with 32 points and almost snatched victory in the final seconds, his attempted buzzer-beater bouncing off the rim as Detroit clung on for a one-point victory.
"We wanted this game," Harris said afterwards. "Obviously, two of the best teams in the East; our group is always up for the challenge. It's a heck of a win for us."
The Pistons-Celtics showdown was one of a packed slate of fixtures as the NBA marked the Martin Luther King Jr. Day holiday in the United States.
Mavericks beat Knicks
Elsewhere in the Eastern Conference on Monday, the third-placed New York Knicks slumped to their fourth straight loss with a surprising 114-97 defeat to the Dallas Mavericks at Madison Square Garden.
Max Christie led the Mavs' scoring with 26 points, 24 of them coming from outside the arc, with Naji Marshall scoring 19 and Cooper Flagg 18.
The Knicks were subjected to boos from their home fans after conceding a massive 75 points in the first half as they slid to defeat.
"We all need to do some soul-searching, some looking in the mirror," Knicks forward Josh Hart said. "Right now we're playing embarrassing basketball. We're not executing on the offensive end. Defensively, we've been abysmal. We've been terrible defensively all year."
Thunder beat Cavs
Shai Gilgeous-Alexander scored 30 points as the Oklahoma City Thunder returned to winning ways with a 136-104 pummeling of the Cleveland Cavaliers.
Two days after suffering an upset defeat by Miami on Saturday, the NBA champions ensured there would be no repeat against Cleveland as the league marked the Martin Luther King Jr. Day holiday with a packed slate of fixtures.
Apart from a brief spell at the start of the first quarter, Oklahoma City led throughout, building up a double-digit lead before accelerating away with a 45-point fourth quarter.
Reigning NBA Most Valuable Player Gilgeous-Alexander's 30-point haul led the way, with center Chet Holmgren providing offensive support with 28 points and Luguentz Dort adding 18 points.
Isaiah Joe chipped in with 16 points from the bench while Aaron Wiggins finished with 12 as five Thunder players recorded double figures en route to a victory that sees Oklahoma City improve to 36-8 at the top of the Western Conference.
Donovan Mitchell led Cleveland with 19 points while four Cavs players – Evan Mobley, Jarrett Allen, Jaylon Tyson, and De’Andre Hunter – all finished with 16 points apiece.
Other games
In Monday's other early game, Giannis Antetokounmpo's 21 points helped the Milwaukee Bucks hold off a late rally in a 112-110 road win over the Atlanta Hawks.
Milwaukee opened a 23-point lead midway through the third quarter, but a stubborn Atlanta line-up chipped away at that advantage and grabbed a slender one-point lead with just over a minute remaining before the Bucks reasserted themselves to dig out victory.
Nickeil Alexander-Walker led Atlanta's scoring with 32 points off the bench, including seven three-pointers, with Jalen Johnson finishing with 28 points for the home side.
The second-placed San Antonio Spurs remain Oklahoma City's closest rivals in the Western Conference and improved to 30-13 on Monday with a comfortable 123-110 defeat of the Utah Jazz (14-29).
French star Victor Wembanyama anchored the Spurs victory with 33 points – including seven three-pointers – on a night when San Antonio's all-around offensive potency was highlighted by seven players finishing in double figures.