The home side were handed an early lead when Antonio Di Natale was brought down in the box inside the opening five minutes, setting the striker up to power the ball in from the penalty spot.
The home keeper continued to come under pressure as the first half progressed, with Marco D'Alessandro testing Meret from range, while a dipping effort from Gaetano Monachello also needed his attention.
Marcelo Estigarribia came close to finding the equaliser shortly before the break for the visitors, but his header from close range ended up in the keeper's arms.
It did not take Atalanta long to find the back of the net after the restart, though, as Monachello hammered the ball in from close range just three minutes into the second half following a quick break.
Atalanta refused to give up, with Estigarribia attempting to set up Monachello for his second of the night, but the striker's finish failed to trouble the keeper.
Udinese quickly charged back down the other end, catching the visitors out with the counter before Adnan crossed the ball in to Di Natale, leaving the captain to fire the ball in on the volley, putting the win beyond doubt for the hosts.