The 2014 U.S. Open champion won, 6-3, 6-2, 7-5, to give Croatia an unassailable 3-1 lead with one singles rubber left to play. Croatia lost that one but it made no difference.
Croatia won its only final 11 years ago and will play Argentina, which defeated Britain earlier Sunday, in the final.
Cilic was Croatia's inspiration at the Kresimir Cosic Hall in Zadar, winning both his singles and teaming up with Ivan Dodig on Saturday to stun top-ranked doubles pair Nicolas Mahut and Pierre-Hugues Herbert.
Gasquet had beaten the big-serving Cilic in two of their three career meetings, but was no competition this time.
Argentina reached its first Davis Cup final since 2011 fter Leonardo Mayer sealed a 3-2 victory over defending champion Britain by beating higher-ranked Dan Evans in Glasgow, Scotland.
Evans would have expected to face Juan Martin del Potro, who beat Andy Murray on Friday, in the decider but the former U.S. Open champion did not want to risk his fragile body after wrist injuries earlier in the year.
The 53rd-ranked Evans still went down, 4-6, 6-3, 6-2, 6-4, to Mayer, who is ranked 114th.
Argentina is through to its fifth Davis Cup final, when it will be looking to win the top team prize in tennis for the first time. It faces Croatia in the final.
Earlier Sunday, a commanding performance from Britain's Andy Murray ensured that the semifinal would go to the deciding match.
Murray, who appeared to strain a muscle in the third set, breezed past Argentina's Guido Pella in straight sets in front of a partisan crowd in Glasgow to win, 6-3, 6-2, 6-3.