Bryan teamed with Ryan Harrison for the first time and helped the Americans outlast Ivan Dodig and Mate Pavic 7-5, 7-6 (6), 1-6, 6-7 (5), 7-6 (5) over almost five hours to cut Croatia's lead to 2-1 in their semifinal on Saturday.
Tiafoe made his opponent work hard in the third before the Croatian won the tiebreaker 7-5 to give his country a solid lead after day one.
The tie continues as Coric takes on Tiafoe in a fifth and deciding rubber.
Croatia will now take on defending champions France in November's final, which will be hosted by the French.
While Coric proved more consistent in the end, Tiafoe's ability to run down shot after shot dragged many points on longer than expected.
Ramos is the same umpire that was accused of being "sexist" by Serena Williams after he handed her a game penalty during her US Open final defeat earlier this month.
"In these later stages of my career I enjoy it so much to play Davis Cup", Cilic said.
"I just hung in there".
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The U.S. had been 2-0 down after Friday's opening singles, but clung on to the hope of reaching their first final in 11 years when 40-year-old Mike Bryan and Ryan Harrison had to fight courageously to win a five-set doubles on Saturday.
Ramos belatedly received public backing from the International Tennis Federation (ITF) and took his place in the chair for Cilic's routine 6-1 6-3 7-6 (7-5) win over Tiafoe.
But Croatia captain Zeljko Krajan is still confident that his team will secure victory in Sunday's first match which sees world number six Cilic face Steve Johnson.
"I didn't know Ramos was sitting in the chair".
Gasquet lost to Albert Ramos-Vinolas 1-6, 6-4, 14/12, before Mahut was defeated by Marcel Granollers 6-7 (2), 6-3, 13/11.
The eighth-ranked Austrian won 6-4, 6-2, 3-6, 6-4 but had to come back twice from a break down in the fourth set.
With his 37th Davis Cup victory, Cilic moved atop Croatia's all-time list, one ahead of Ivan Ljubicic.