Udinese had pace on the counter but fell behind after losing concentration in first-half stoppage time, with Denis stooping to open his account for the campaign with a diving header.
The second half then began with a controversial incident as Udinese were awarded a penalty by the referee, only for that decision to be reversed after lengthy discussions involving the linesman.
Francesco Guidolin was furious and his mood was not improved as Atalanta doubled their advantage soon after with a well-worked second goal.
Denis applied the finish, sprinting through to meet Luca Cigarini's chipped pass before unleashing a composed finish into the corner.
The visitors were unable to mount a response in the closing stages, as Atalanta closed out their first victory in four attempts.