The 21-year-old Coric came from two sets to one down to defeat the 20-year-old debutant Tiafoe 6-7 (7-0) 6-1 6-7 (13-11) 6-1 6-3 and finally subdue a remarkable fightback from the Americans, who had earlier levelled the tie 2-2 from two down.
Late understudy Sam Querrey had rewarded captain Jim Courier's gamble by taking the tie into a decider in the fourth rubber, battling back from two sets down to beat home talisman Marin Cilic 6-7 (7-2) 7-6 (8-6) 6-3 6-4.
Yet with everything resting on the first duel between two of tennis's brightest young stars Coric overcame the loss of two very contrasting tiebreaks to book Croatia's progression.
After Coric had capitulated 7-0 in the first-set breaker and 13-11 in the third set decider, home fans feared Croatia could be on the wrong end of one of the competition's unlikeliest comebacks.
Coric, though, held his nerve against the equally brilliant young American and finally prevailed after four hours six minutes.
At 2-0 down after Friday's singles, there had not seemed the remotest chance of a US team without their top two players John Isner and Jack Sock recovering to reach a first final in 11 years.
Only when 40-year-old Mike Bryan and Ryan Harrison clawed out a five-set doubles win on Saturday, were the Americans given even faint hope.
But Courier was rewarded for resting his No.1 Steve Johnson and instead threw Querrey into the must-win fourth rubber even though the 30-year-old had never previously beaten Cilic in six attempts.
World No.6 Cilic was found wanting when, after winning the opening set on a tiebreak, he stood on the verge of a commanding lead.
He took a 6-1 lead in the second set tiebreak in superb fashion only to crumble under Querrey's onslaught with the American world No.61 winning seven straight points.
Cilic's early dominance then evaporated as Querrey took the next two sets to win in three hours and eight minutes.