The match was umpired by Portugal's Carlos Ramos -- the official accused of sexism by American Serena Williams after docking her a game during a tumultuous U.S. Open final.
Thankfully for Ramos this match was a more tranquil affair, apart from the noisy Croatian crowd cheering on their man.
Cilic's victory made him Croatia's most successful Davis Cup player, moving him past the 36 rubbers won by Ivan Ljubicic.
Earlier Borna gave Croatia a flying start by defeating Steve Johnson 6-4 7-6(4) 6-3 in the opening rubber.
The 21-year-old Coric had lost his only previous encounter with the U.S. number three at Roland Garros but on Zadar's clay courts drew first blood for the hosts against Johnson, who was leading his country in the absence of John Isner and Jack Sock.
Coric's emphatic win just underlined why the Croats, who had won all their four previous ties with the Davis Cup's most successful nation, were overwhelming favourites to repeat the dose and reach the final of the last Davis Cup to be played in its traditional historic format before next year's revamp.
World number 18 Coric was impressive from the start, taking an early break for 2-1, aided by Johnson missing plenty of first serves, and controlling the 35-minute opening stanza.
The youngster brought the crowd to their feet with one extraordinary 'tweener' lob that left Johnson flat-footed as he earned another quick break in the second set but the American fought back strongly to take it to a tiebreak.
Coric again stepped up his game, reeling off the final four points of the breaker, featuring a glorious forehand winner and a fine volley, to seal it 7-4.
He then forged home his advantage over the world number 30 in the third set to wrap up victory in two hours and 24 minutes.
(Editing by Christian Radnedge and Toby Davis)
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