The Colombian substitute beat goalkeeper Pietro Terracciano at his near post with a powerful effort in the dying minutes to clinch victory, despite a poor Juve performance in which they failed to muster a shot on target until the 86th minute.
Fiorentina defender Nikola Milenkovic's dismissal in the 73rd minute injected some life into the home side as Federico Chiesa blasted a shot off the crossbar and Alvaro Morata had a goal ruled out for offside before Cuadrado broke through at the death.
"We are happy with the win, we will continue on this line," Cuadrado told DAZN.
"We are a team and a family, we have to fight for our team-mates until the end."
It was coach Massimiliano Allegri's 200th win as Juventus coach in all competitions, making him only the third manager to achieve the feat since 1929/30 after Giovanni Trapattoni (319 wins) and Marcello Lippi (227).
But the 54-year-old's side did little to suggest an imminent transformation in form after an underwhelming display.
The result leaves eighth-placed Juventus on 18 points, level with Fiorentina above them.
Back-to-back defeats against midtable Sassuolo and Hellas Verona left Juventus in crisis ahead of the visit of their rivals from Florence, although a midweek Champions League win over Zenit St Petersburg was a positive sign.
There was no obvious improvement on display from the hosts in the first half, though, as Fiorentina created the best chance of a dull start to the game when Riccardo Saponara blasted over from 12 yards.
Milenkovic's expulsion proved to be the turning point, as the Serbian was shown a second yellow for a clumsy challenge on Chiesa and Juve pressed forward for the final quarter of the match.
Chiesa smashed a half-volley off the underside of the bar and Morata tucked in a cross only for it to be ruled out for offside, and the Old Lady finally registered their first shot on target when Chiesa's shot was beaten away by Terracciano with four minutes of regulation time remaining.
Cuadrado, who was introduced with 11 minutes remaining, made the difference with a piece of superb individual skill, skipping past Cristiano Biraghi and drilling a low shot home.