The first chance of the clash fell to Nico Lopez, whose half-volley was deflected behind by one of his own teammates to the relief of the six-time winners.
Moments later, winger Juan Vargas unleashed a powerful drive from the right-hand side of the box, which drifted wide of the target.
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The deadlock was broken on the 26 minute mark, as the Viola's sloppy play inside the Udinese box was pounced on by Roberto Pereyra, who made a run towards the opposition box before feeding Antonio Di Natale to leave the forward with a tap-in.
The visitors felt they should have had a penalty shortly after after when the ball appeared to strike Dusan Basta on the hand, but the referee felt otherwise and waved play on.
Alessandro Matri was the next Fiorentina player to come close to bringing his side level having found his way in on goal, though Scuffet read the danger and dived at the ball to end the attack.
Vicenzo Montella's men did make the breakthrough before the interval, however, courtesy of Vargas who dodged a Di Natale tackle, before striking the ball first time from 25-yards to even the score.
Fiorentina continued to push for a second, and nearly got their reward when Joaquin picked out Vargas with a fine cross, though the midfielder couldn't keep his attempt on target.
The side sitting fourth in the Serie A table would live to rue the missed opportunity, as with less than 10 minutes remaining, Luis Muriel fired the hosts back into the lead with a drive from distance to give Udinese the advantage ahead of the ties second leg.