Both players had their chances but it was Dodig who found the breakthrough at the third time of asking.
But Dodig hit back with his first break of the match and the set eventually went to a tie-break.
Fabio Fognini won the deciding match for Italy against Ivan Dodig, coming from behind to triumph 4-6 6-4 6-4 6-4.
Meanwhile in Hyogo, Croatia lead Japan 2-1 after Ivo Karlovic and Ivan Dodig beat Tatsuma Ito and Yuichi Sugita 6-4, 6-4, 3-6, 6-3.
On the men's ATP Tour, Ivan Ljubicic slipped to a 6-0 6-3 defeat to fellow Croatian Ivan Dodig at the Monte Carlo Masters Sunday.
But he made no mistake in Dodig's next service game, creating two more chances and taking the second with a drive volley behind his opponent.
Italy hold the edge in their tie in Turin after Simone Bolelli and Fabio Fognini won their doubles match against Marin Cilic and Ivan Dodig on Saturday.
Dodig continued to trade blows with his more illustrious opponent in the second set but he failed to capitalise on the chances he created and Djokovic struck to establish a 5-3 lead.
The 2008 finalist and seventh seed, Jo-Wilfried Tsonga demolished Slovakia's Blaz Kavcic 6-2 6-1 6-4 to set up a fourth round meeting with compatriot Richard Gasquet who also completed a straight-sets victory (6-3 7-6 6-0) over Ivan Dodig.
Federer, who said before the start of the tournament that he was "excited" by the return of Nadal to top flight action, was playing Ivan Dodig of Croatia in a later third round match as he defends his title.
Defending champion Federer, who said before the start of the tournament that he was "excited" by the return of Nadal to top flight action, thrashed Croatia's Ivan Dodig 6-3 6-1 to set up a fourth-round clash with his friend and fellow Swiss Stanislas Wawrinka .
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