The visitors took the lead in the 23rd minute when Cuadrado latched onto Joaquin's cross inside the box and diverted the ball into the far corner.
Fiorentina added a second nine minutes later when Ilicic's curling effort took a deflection before looping over the head of Curci and into the net.
Ilicic almost added a third before half time, but he saw his powerful effort beaten away by the keeper.
After the break Cuadrado was close to scoring his second of the match but after working some room in the box, he saw his strike saved by Curci.
The Viola continued to look the more likely team to score the next goal and Nenad Tomovic wasted a good opening when blazed a shot wide from close range.
With three minutes left to play, Cuadrado sealed the win with a powerful strike into the corner from long range, as the visitors strengthened their position of fourth place in the Serie A table.