Atalanta finished the match with 10 men when German Denis was sent off in the dying seconds for a late challenge on Pizarro, but the game was beyond them by that stage.
Guglielmo Stendardo was deemed guilty of handling a Adem Ljajic flick in the box and Pizarro sent the keeper the wrong way for the opening goal on 61 minutes.
Substitute Marcelo Larrondo raced on to a Juan Cuadrado through ball to smash a powerful second goal into the net 11 minutes later to put the game beyond the home side.
Fiorentina close to within three points of third-placed Milan and are now four points ahead of Lazio.
Vincenzo Montella's side were missing several first teamers, with Stevan Jovetic, Luca Toni, Stefan Savic, Mati Fernandez and Rafal Wolski injured, plus Nenad Tomovic suspended.
Atalanta started brightly as both sides traded attempts but Montella's decision to bring on the partly-fit Adem Ljajic paid off as the Serbian midfielder won a penalty when his flick struck Stendardo's hand.
Pizarro converted the spot-kick to put La Viola ahead, and it should have been two when they were denied a clearer-cut penalty as another Ljajic effort his the arm of Stefano Lucchini this time.
They did get a second and it came in some style when Larrondo hammered a left-foot strike past the helpless Consigli, the ball flying in at the near post from a tight angle.
Ljajic almost added a third when his effort was saved by Andrea Consigli and things went from bad to worse for the hosts when Denis was dismissed for a terrible tackle on Pizarro.
It was Atalanta's 13th red card of the season, the worst record in Serie A.