After rain delays of six and 30 minutes, Nadal fought off two set points in the first-set tiebreaker. Play was stopped for 1 hour, 56 minutes with Nadal leading 2-0 in the second set.
"All the matches are difficult here," Nadal said. "It was a tough first set. Then in the second, I was able to take advantage at the beginning, but then he broke me back."
Cilic topped Australia's John Millman 6-3, 6-4.
Poland's Hubert Hurkacz upset fourth-seeded Stefanos Tsitsipas of Greece 6-4, 3-6, 6-3, France's Richard Gasquet beat fifth-seeded Kei Nishikori of Japan 6-7 (6), 6-2, 7-6 (4); France's Adrian Mannarino, topped 11th-seeded Borna Coric of Croatia, 6-2, 6-1; and Christian Garin of Chile knocked off 12th-seeded American John Isner 6-3, 6-4.
Sixth-seeded Karen Khachanov of Russia edged Stan Wawrinka of Switzerland 6-4, 6-7 (3), 6-2; seventh-seeded Fabio Fognini beat American qualifier Tommy Paul 7-6 (3), 6-3; eighth-seeded Daniil Medvedev of Russia topped Kyle Edmund of Britain 6-3, 6-0; and 10th-seeded Roberto Bautista Agut of Spain edged Diego Schwartzman of Argentina 6-2, 7-5.