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Udinese hit back within two minutes as Antonio Di Natale reacted quickly to fire home a loose ball inside the box and bring his side level.
Guilherme then had a glimpse of goal for the visitors when he was found at the back post by Di Natale, but he could not direct his header on target.
However, Torino's reprieve was short lived as Panagiotis Kone nodded the ball in from close range from the resulting corner to give Udinese the lead.
Quagliarella saw a powerful effort saved and Maxi Lopez dragged a shot wide as the visitors pushed for an equaliser.
Torino came close again just before half time when Quagliarella smashed a volley from a tight angle against the post.
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Udinese held onto their lead going into the break and, within four minutes of the second half, they scored a third as Wague powered in a header from Di Natale's corner.
The game continued to be played at a frantic pace and Lopez almost pulled a goal back for the visitors when he was slipped through and unleashed a low shot that was blocked by the legs of Orestis Karnezis.
Torino did manage to score their second of the afternoon in the 69th minute when Marco Benassi's deflected strike found the net.
As the game entered the final 10 minutes, substitute Amauri almost scored with a spectacular overhead kick that was well held by Karnezis.
With three minutes left to play, Omar El Kaddouri sent an effort narrowly wide and that was as close as Torino came to finding an equaliser as they fell to their first defeat in 12 league games.