The victory was Cilic's 21st straight against an American dating to 2008. The Croatian has reached three straight finals for the first time in his career. He beat Tommy Haas to win at Zagreb earlier this month, and then lost to Tomas Berdych at Rotterdaam last week.
"It's probably the best three weeks I've played in my life," said Cilic, who reached his 20th career ATP final. "I couldn't be happier. I'm enjoying these moments."
The start of the night match was delayed nearly three hours because of rain at the Delray Beach Tennis Center and began at 11:25 p.m.
Cilic will face Kevin Anderson of South Africa in the final on Sunday. Anderson, who won this event in 2012, advanced with a 6-2, 6-4 victory over American qualifier Steve Johnson. Anderson, seeded No. 4 and ranked No. 21, had 13 aces and won 86 percent of his first-serve points.
"I'm pleased with the way I executed that part of my game," said Anderson, who has a 7-2 record this year. He was 0-3 in finals last year.
Anderson played near-flawless tennis in the first set against Johnson, who admitted having some nerves playing in his first ATP semifinal. Anderson broke serve twice to take a 4-0 lead. Johnson, meanwhile, struggled some with his first serves in the first set.
"Getting that first break was really key for me," Anderson said. "I was hitting my spots."
Johnson, who is expected to move up in the rankings to about 114 after entered this tournament at No. 142, settled down in the second set. But with the set tied 3-3, Anderson broke Johnson's serve and then closed out the match on his next two serve opportunities.
Johnson, a former two-time NCAA singles champion from the University of Southern California, beat the top-seeded Haas and sixth-seeded Feliciano Lopez to reach the semis,.
"I definitely was a little uptight in the beginning, but credit (Anderson). He played very well," Johnson said. "He was a better tennis player today, and I have nothing to hang my head about."
Brothers Bob and Mike Bryan, the No. 1 doubles team in the world, easily defeated fellow Americans Ryan Harrison and Jack Sock 6-2, 6-1 in the doubles semifinals. Their opponent in Sunday's final will be Czechs Frantisek Cermak and Russian Mikhail Elgin, who defeated Samuel Groth of Australia and Max Mirnyi of Belarus 6-4, 3-6, 10-8.