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Udinese were in control from the outset as Roberto Pereyra and Maicousel tested Gianluca Cerci in the Bologna goal.
The home side continued to press before the break and were only denied by the crossbar from Andrea Lazzari's powerful strike.
On 65 minutes Udinese were awarded a penalty when Pereya was brought down by a late challenge from Saphir Taider.
Di Natale stepped up but his low effort was denied by a save at full stretch from Cerci.
Luis Muriel was next to test Cerci with an overhead kick, which needed a late save to be tipped over the bar.
Bologna saw out the final few minutes to earn a battling point away from home and frustrate Francesco Guidolin's side.