Cilic's 6-3, 6-3, 4-6, 6-2 win gave Croatia an insurmountable 3-1 lead in the best-of-five quarterfinal competition.
After Cilic's clinching win, the U.S. team's Bob Bryan, who won Saturday's doubles with his brother Mike, ended the matches with a 5-7, 6-3, 7-6 (7-4) win over Roko Karanusic, setting the overall final score at 3-2.
Cilic had earlier defeated American Mardy Fish. Blake's loss was his second of the weekend, having fallen Saturday to Ivo Kalovic.
Croatia will clash with the Czech Republic team in the semis in September, The Czechs advanced Sunday with a 3-2 win over Argentina.
The Croatian team is in the semifinal round for the first time since it won the cup in 2005.