New Orleans has won just eight times in 30 tries against teams from the Western Conference, but improved to 13-5 against teams from the East.
The Pelicans trailed by four to open the fourth period, but took the lead for good with 4:53 to go on a 3-point play by Eric Gordon. Two free throws by Austin Rivers with 16 seconds remaining gave New Orleans a seven-point cushion.
Davis made 9-of-14 from the field and 9-of-10 from the foul line. Brian Roberts added 19 points for the Pelicans and Gordon had 18.
Atlanta, which lost its second straight, was led by Paul Millsap with 26 points and 10 rebounds.