Borna Coric won the deciding singles match with Frances Tiafoe to send Croatia through to a Davis Cup final against reigning champions France.
Sam Querrey's victory over Marin Cilic earlier in the day levelled the tie at 2-2 but Croatia, who were 2-0 up after Friday's singles, were able to reach their second final in three years as Coric beat Tiafoe in five sets.
Davis Cup debutant Tiafoe had lost in straight sets to Cilic on Friday, but he was one set away from a victory against Coric, only for the American to win just four matches across the final two.
Coric sealed a 6-7 (0-7) 6-1 6-7 (11-13) 6-1 6-3 success in over four hours as he improved his record to 3-0 in deciding rubbers at the Davis Cup.
Steve Johnson had been due to face Cilic at the start of the day, but Querrey was drafted in to tackle an opponent who had won each of their previous six meetings.
Cilic won the opener following a tie-break, but squandered five set points in the second breaker to allow Querrey to level.
The American then broke his opponent three times across the final two sets to claim a 6-7 (2-7) 7-6 (8-6) 6-3 6-4 success that forced a deciding fifth match in the tie.
At 1-1 in that contest, Coric was in control when 5-1 up in the third set, but the Americans were dreaming of a come-from-behind victory when Tiafoe rallied to win that set in a breaker.
Yet Coric forced a one-set shoot-out and held his nerve in Zadar to send his country through to a final with France in November.
"This is the most special moment of my whole life...by far," said Coric.