Torino dominated the early exchanges and the hosts were guilty of missing a good chance to break the deadlock when Cristian Molinaro fired just wide from the edge of the box.
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As Udinese's first-half struggles continued, they were fortunate to remain on level terms heading into the break after Giuseppe Vives rattled the post from close range.
Orestis Karnezis was forced into action just after the interval to get a crucial hand on a swerving effort from Vives.
Vives continued to be an influential figure, and it was his inviting cross which produced the winning goal in the 62nd minute as Quagliarella took full advantage of poor marking to head home.
Despite eventually applying pressure late on, Udinese's below-par attacking play would ultimately cost them as Torino held on for just their second Serie A win of the season.