The match in Udine had been pushed back from 7 December after heavy rain made the pitch unplayable.
Gian Piero Gasperini's side also dropped points in its goalless draw against Genoa last weekend.
The Bergamo side stays sixth, 10 points behind leaders AC Milan, and one point adrift of third-placed Napoli and Roma, who occupy the final two Champions League places.
Champion Juventus is fifth equal on 33 points with Atalanta.
Roberto Pereyra opened the scoring after just 23 seconds at the Dacia Arena surprising goalkeeper Pierluigi Gollini from a tight angle after shaking off Atalanta defensive duo Jose Luis Palomino and Cristian Romero.
Udinese could thank goalkeeper Juan Musso for some fine saves denying Joakim Maehle and Rafael Toloi before Colombian Luis Muriel levelled for Atalanta just before the break for his 11th goal this season.
Gian Piero Gasperini brought Duvan Zapata, Josip Ilicic and Robin Gosens off the bench to push for the three points but the hosts held on for a point.
Udinese, in 15th, stretched its winless run to eight games.
It next plays second-placed Inter Milan at home this weekend.