With Detroit down 81-78, Billups hit four straight free throws to put the Pistons ahead with 2:27 to go.
After Ben Wallace blocked a Mike Miller shot, Billups hit a fast-break 3-pointer, then hit another after a Pau Gasol turnover. He then missed a free throw that could have made it a four-point play with 1:28 left.
At 36-5, Detroit has the fourth-best 41-game record in NBA history, trailing only the 38-3 starts by the 1995-96 Bulls and 1971-72 Lakers, and the 37-4 record of the 1966-67 76ers.
Billups finished with a game-high 34 points, Richard Hamilton added 19 and Tayshaun Prince scored 16 for the Pistons.
Gasol led the Grizzlies with 31 points and Miller had 20.
Detroit allowed the Grizzlies to shoot 50 percent from the floor in the first half, but stayed in the game by outrebounding Memphis 21-12. Gasol scored 18 points to help the Grizzlies grab a 46-44 lead at the break.
The Pistons led for much of the third quarter, but the Grizzlies rallied and Gasol hit a long 3-pointer at the buzzer to give Memphis a 68-64 advantage.
Detroit was within 79-78 when coach Flip Saunders was called for a technical foul with 3:22 to go, and the Pistons picked up another technical for a second delay-of-game warning 17 seconds later. Mike Miller made both shots, but Billups scored the next 10 points.
Notes:@ Memphis rookie Anthony Roberson, who grew up about an hour north of the Palace, was inactive. … Detroit needed double overtime to win 106-104 in Memphis on Dec. 19. … The Pistons are only the third team in 20 years to start the same five players in their first 41 games, joining the 1999-2000 Indiana Pacers and 2002-03 Golden State Warriors. … The Pistons need to go 28-13 in the second half of the season to break the franchise record of 63 wins in a season. … The 1991-92 and 1996-97 Bulls also started 36-5.
