The Crown Estate said it was "the most significant milestone yet" in its Round Three programme.
The format for Round Three was supposed to be the same as that of the First Debate.
Belgian eigth seed Kristof Vliegen beat Switzerland's Marco Chiudinell 7-6 (7-4) 6-0 to progress to round three.
Carrick Rangers, Donegal Celtic, Institute and Ballymena United also made it through to the round three draw.
Serbia's Jelena Jankovic won through to round three after a 6-3 3-6 6-1 win over American Bethanie Mattek-Sands.
McIlroy started the final round three shots behind overnight leader Alvaro Quiros, who faded to a three-over-par 73.
And the six-times Grand Slam winner, who plays India's Sania Mirza in round three, is certain she can improve.
Moore continued to pepper Piccirillo in round three, before knocking him out with a huge left to the body.
The two-time quarterfinalist will face Russian 27th seed Mikhail Youzhny in round three.
But Serena Williams beat Vera Dushevina 6-7 7-6 7-6 to reach round three.
Fehintola started well but went down for the first time from a left hook to the temple in round three.
The Scot began the final round three shots back but pulled himself into contention with three successive birdies from the second.
Over the past 10 years the Crown Estate has held five offshore wind leasing rounds, the most recent being Round Three.
Cilic was joined in round three by French fifth seed Jo-Wilfried Tsonga and Spanish sixth seed Fernando Verdasco, who both recorded straight-sets victories.
Russia's Svetlana Kuznetsova , seeded 14, made it safely through to round three after a 7-5 6-3 victory over Spaniard Maria Jose Martinez-Sanchez.
The 21-year-old will play another surprise winner, Latvia's Anastasija Sevastova, in round three after her 6-4 6-2 win over Belgium's 21st seed Yanina Wickmayer.
World number one Serena Williams also overcame Swiss opposition to join her sister in round three after a 7-6 6-1 win over Timea Bacsinszky.
And though Cilic twice went a break down in the deciding set, he held on to claim it 6-4 and progress to round three.
Williams will now play Shahar Peer in round three after the Israeli recovered from dropping the opening set to beat Polona Hercog 2-6 7-5 6-3.
The winner of that match will play either Canadian 23rd seed Milos Raonic or Germany's Philipp Petzschner in round three, with Djokovic looming after that.
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The No. 9 seed Juan Martin del Potro booked a place in round three with a 6-7 7-6 6-4, 6-4 win over France's Edouard Roger-Vasselin.
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Oosthuizen led by five shots after round three and withstood another breezy day to take a four-shot lead over Casey into Sunday's finale on the Old Course.
Woods, the highest-placed American in the field, was seven shots off the pace after round three and was unable to close the gap with a modest two-under 71.
Unlike Federer, the world No. 3, Britain's Andy Murray , was made to work hard for his place in round three after overcoming Germany's Florian Mayer 7-6 7-6.
Isner, best known for his marathon encounter with Nicolas Mahut at this year's Wimbledon, eventually triumphed 4-6 7-6 (11-9) 7-6 (9-7) and will face Roger Federer in round three.
Sharapova will next face Estonian 23rd seed Kaia Kanepi, who upset former world No. 1 Caroline Wozniacki in round three and then beat unseeded Dutchwoman Arantxa Rus on Monday.
"Maybe I did, but I don't remember, " said the Swede, who will play Albert Montanes in round three after the 29th seed waltzed past Tobias Kamke 6-3 6-2 6-1.
Czech 29th seed Radek Stepanek crashed out 7-5 7-5 6-3 against French world No. 81 Nicolas Mahut, who will likely face Djokovic in round three if he overcomes Japan's Tatsuma Ito.
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Federer is on course for a semifinal clash with the world's top ranked player after Djokovic eased into round three with a 6-0 6-4 6-4 win over Blaz Kavcic of Slovenia.
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