The loss was Boston's first at home this season having won their first 12 home games.
The Pistons held off a late Boston rally and won the game when Billups converted his free throws after being fouled by Tony Allen.
"It was playoff-intense, pretty aggressive out there," Detroit coach Flip Saunders told reporters. "I would say, by far, it was the most aggressive game this season, with all five guys going against each other.
"It got a little bit chippy, so it was a little bit of a playoff-like atmosphere."
Detroit led 85-79 with 3:34 to play but Eddie House and Ray Allen hit three-pointers to tie the game. The score was level until the final seconds, when Paul Pierce missed a fadeaway that would have given the Celtics the lead.
Rasheed Wallace grabbed the rebound and called a timeout with 1.7 seconds left. On the inbound, Allen fouled Billups, leading to the winning free throws.
Billups had a game-high 28 points, including 12 in the fourth quarter, and added eight assists. Richard Hamilton added 21 points and Wallace nine points and 13 rebounds for the Pistons.
Kevin Garnett led Boston with 26 points and 12 rebounds and Ray Allen added 24 points. Despite being the losing team, Boston had 49 percent possession to Detroit's 40 percent.
"It's a good lesson for our guys," Boston coach Doc Rivers said.
Detroit (18-7), the Central Division leaders, won their third successive game.
Dirk Nowitzki scored 31 points when the Dallas Mavericks hung on to edge ahead of the Phoenix Suns 108-105 in the latest showdown between the two Western Conference rivals.
The New York Knicks defeated the Cleveland Cavaliers 108-90 with LeBron James grabbing 32 points, eight rebounds and six assists for the Cavaliers, who have lost nine of their last 11 games.
The Atlanta Hawks outlasted the slumping Miami Heat to win 117-111 in overtime. The Heat have been defeated in eight of their past 11 games and lost Alonzo Mourning due to a knee injury.
The Charlotte Bobcats rallied from a 12-point deficit with six minutes left to beat the Utah Jazz 98-92. Gerald Wallace had 26 points to lead Charlotte.
The San Antonio Spurs lost their third successive road game when Rudy Gay hit a three-pointer over Tim Duncan as the final buzzer sounded, lifting the Memphis Grizzlies past the Spurs 88-85.
Dwight Howard had 21 points, 11 rebounds and three blocks when the Orlando Magic beat the Houston Rockets 97-92. Yao Ming had 19 points and 17 rebounds for the Rockets, who lost Tracy McGrady with a knee injury after the first half.