On the quarter-hour mark, Nahuel Valentini came close with his head, but his effort was blocked.
Against the run of play, the Nerazzurri took the lead midway through the first half when Giuseppe De Luca put the ball in the back of the net.
Greco attempted to pull the away side back just two minutes later, but he was off target.
After a slow start to the second half, Livorno threatened again, but Marco Biagianti's strike in the 56th minute failed to hit the target.
He would come to regret that missed opportunity, because Atalanta doubled their lead just three minutes later, thanks to a strike by German Denis.
Innocent Emeghara attempted to pull a goal back with an effort in the 75th minute, but failed to hit the target.