Detroit won the contest despite a 67-61 deficit going into the final period. Its deficit was 52-45 at the half.
Billups finished with 28 points and eight assists, including sinking both free throws with 1/10th of a second remaining following a foul by Boston's Tony Allen.
Detroit's Richard Hamilton added 21 points. Antonio McDyess had 13 points and 10 rebounds, while Rasheed Wallace had nine points and 13 rebounds for the Pistons, who have won four in a row at Boston's Fleet Center.
Boston was led by Kevin Garnett's 26 points and 12 rebounds. Ray Allen had 24 points and five rebounds.
Rajon Rondo had 14 points and seven assists, and Paul Pierce added 11 points, four rebounds and four assists for the Celtics, who had their nine-game winning streak snapped.
They have not won 10 in a row since March 11-April 4, 1986.