The visitors took the lead in controversial circumstances on 17 minutes when Davide Astori's header bounced down off the crossbar and onto the line.
There was some confusion following the incident as the Udinese players protested that the whole of the ball did not cross the line, but the referee eventually blew his whistle to give Roma the goal.
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Maicon's low drive from distance was pushed away by the Greek before Iturbe and Emanuelson were both frustrated by the shotstopper.
Karnezis came to his team's rescue again deep in added time, saving from Alessandro Florenzi after the midfielder was put clear through on goal.
Roma were able to hold on for the victory following some late pressure from Udinese, but their afternoon was soured during the final 10 minutes when Adem Ljajic was stretchered off the pitch with a knee injury.