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|Australia's Nick Kyrgios during the ATP Masters tournament tennis match at the Foro Italico in Rome. (Andreas SOLARO/AFP)|
However, Australian firebrand Nick Kyrgios was disqualified after a foul-mouthed rant during his third-round match against Norway's Casper Ruud which could land him with a lengthy suspension.
Federer came through 2-6, 6-4, 7-6 (9/7) against 13th seed Coric after earlier beating Portugal's Joao Sousa 6-4, 6-3 in the second round as players doubled up after rain wiped out play on Wednesday.
Federer, a four-time finalist in Rome, trailed 0-4 before dropping the first set, coming back in the second and saving two match points in the third set tie-break.
"Things definitely went my way tonight," said Federer.
Nadal dropped just two games throughout the day, dispatching Georgian Nikoloz Basilashvili 6-1, 6-0 in the third round after earlier also dropping just one game against France's Jeremy Chardy.
"I have been improving every week," said Nadal, who next plays fellow Spaniard Fernando Verdasco as he targets a first title on clay this season before his French Open defence.
"It's been a peculiar clay court season for me. I had some tough period of time. Has been little bit difficult to hold the body and the tennis level for every day. I just try. I have another chance tomorrow."
Djokovic, a four-time Italian Open winner, next plays Argentine seventh seed Juan Martin Del Potro for a place in the semi-finals.
Djokovic beat German Philipp Kohlschreiber 6-3, 6-0 after earlier easing past Canadian Denis Shapovalov 6-1, 6-3.
In the women's event, Japan's Naomi Osaka, the top seed, also won twice to retain the world number one ranking ahead of the start of the French Open.
She eased into the last eight with a 6-3, 6-3 win over Mihaela Buzarnescu after earlier beating Slovak Dominika Cibulkova by the same scoreline.
Osaka was at risk of losing her world number one ranking to Simona Halep but the Romanian was knocked out by Czech teenager Marketa Vondrousova losing 2-6, 7-5, 6-3.
Halep's defeat also guarantees Osaka's top seed at the French Open.
SWEARING, CHAIR THROWING
Kyrgios's tournament finished early after he threw a chair across court in a tantrum, before storming off and being disqualified.
The controversial 24-year-old, who had marked his presence in Rome by criticising Djokovic and Nadal in an interview on Wednesday, started arguing with the umpire in the third set.
World number 36 Kyrgios had levelled the match at one set all but was given a game penalty early on in the third set for swearing.
His response was to kick out at a water bottle before throwing a chair onto the court, packing his bag and then storming off court, shouting I am f***ing (expletives) done''
Kyrgios was automatically disqualified with Ruud winning 6-3, 6-7 (5/7), 2-1 to advance to the third round before losing to del Potro.
"I think he got what he deserved," said Ruud of Kyrgios who was then fined 20,000 euros and made to forfeit his tournament winnings.
"He thinks he can do what he likes."
Kyrgios had already been at the centre of controversy during Wednesday's washout when he told the NCR Tennis Podcast that Djokovic had "a sick obsession with wanting to be liked" and that the Serb's post-match celebration was "cringeworthy".
Nadal was described by the Australian as "super-salty".
Fifth seed Dominic Thiem slammed tournament organisers after he was dumped out 4-6, 6-4, 7-5 by Verdasco.
Austrian Thiem complained competitors were left hanging around during the rain-impacted day 24 hours earlier.
"I really dislike how we players get treated at this tournament because yesterday was, in my opinion, not acceptable," said last year's French Open runner-up.
"I'm quite pissed about it. I was tired, exhausted, today because of all these shitty things," he added.
Japan's Kei Nishikori, the sixth seed, got past American Taylor Fritz 6-2, 6-4 and then Jan-Lennard Struff of Germany in three sets.
Czech second seed Petra Kvitova eased past Yulia Putintseva of Kazakhstan 6-0, 6-1 in her second round match.