Cyril Thereau thought that he had given the visitors the lead early on, but his effort was ruled out for an offside.
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It was Verona who then came within inches of breaking the deadlock as Panagiotis Tachtsidis crashed a powerful effort from just outside the box against the bar.
Bosko Jankovic was next to threaten with a curling effort that he placed narrowly over the top.
The hosts were then dealt a blow on the hour mark as Jacopo Sala was shown a straight red card for a challenge on Thereau.
Udinese immediately made their numerical advantage count as Di Natale met Silvan Widmer's cross and flicked the ball beyond the keeper to give his side the lead.
Thereau almost scored a second for the visitors, but his low effort smashed against the outside of the post.
Luca Toni came close to grabbing an equaliser for the hosts with a strike that he curled narrowly over the top, but they were unable to find the net, and they finished the match with nine men following Rafael Marquez's late sending off.