Billups, a 92% free-throw shooter, missed a pair with seven seconds left.
Davis, shooting only six for 21 to that point, took a pass from P.J. Brown, went past Ben Wallace and dunked over Bob Sura.
"We had a great play drawn up that we practiced in shoot-around," Davis said. "When I came off the pick, I saw Ben leaning and I knew I had to get to the hole. There were no jumpers being shot -- I was missing those all night. I had to get to the hole."
Jamaal Magloire led New Orleans with 20 points and 15 rebounds.
Milwaukee 82, Toronto 62 -- Toni Kukoc had 15 points at Toronto to help the Bucks
end a three-game losing streak.
Vince Carter had 15 points for the Raptors, who scored fewer than 70 points for the fourth time this season.
Juan Dixon and Kwame Brown had 17 points each for Washington.
Golden State 78, Portland 72 -- Erick Dampier had 11 points and 17 rebounds at Oakland for his seventh straight double-double to help the Warriors end a five-game slide.
Zach Randolph had 17 points for Portland, winless in five road games.
Sacramento 110, Chicago 99 -- Peja Stojakovic had 21 points and Anthony Peeler scored 15 of his season-high 18 in a decisive 28-7 second-quarter run at Sacramento as the Kings won their fifth in a row.