Cilic beat Fish 4-6, 6-3, 6-7 (3), 6-1, 8-6 in a match that lasted 4 hours, 23 minutes and ended at 1 a.m. local time.
"His serve is probably the best on the tour," Blake said. "When he hits his spots, there is nothing we can do about it."
The best-of-five series resumes Saturday with doubles and concludes with reverse singles Sunday. The world's No. 2 doubles team, the Bryan twins, will represent the U.S.
"If we win doubles, as expected, we certainly have a chance" for an overall victory, U.S. captain Patrick McEnroe said.
The U.S., a 32-time champion, has lost to Croatia the two previous times they met in Davis Cup -- in 2002 and in 2005 when Croatia won the competition. The U.S., the 2007 champion, is bidding to reach the semifinals for the fourth straight year. The winner will face the Czech Republic or Argentina in the semifinals in September.
In Friday's other quarterfinals, Israel took a surprising 2-0 lead at home against Russia; Germany and host Spain were tied 1-1; and Argentina was even at 1-1 at the Czech Republic.
Cilic converted the first of his three match points on Fish's serve in the fifth set with a powerful forehand down the line. Cilic served for the match at 5-3, but allowed a break that extended the match.
"It was tough, but we did it," Cilic said.
"I certainly gave everything I had," Fish said. "Clay is not my best surface, but I think I played very well."
Karlovic, ranked 27th, won a five-set match for the first time after having lost 11 straight five-setters.
"Playing five sets for me, being tall and heavy, is not easy," he said. "This is one of the biggest wins in my career."
Blake, ranked 17th, seemed to have the match under control after taking the first two sets. But Karlovic persevered, converting his first break point with an inside-out forehand in the third set for a 2-0 lead. Then he had two perfect volleys and a forehand return in the fourth-set tiebreaker, which he won 7-3.
"The tiebreak in the fourth set is the best I have seen him play," said Blake, adding that Karlovic's serve "sometimes makes us looks silly."
Karlovic won on his second match point after a long volley by Blake.
"My job was to win that match," Blake said, "and I didn't do the job."