Marin Cilic, the 2014 U.S. Open champion, beat John Isner 7-6(11) 6-3 6-4 in a battle of big servers to set up a deciding rubber between 19-year-old Coric and Jack Sock.
The 54th-ranked Coric prevailed 6-4 3-6 6-3 6-4.
"Our team just shows how much spirit we have, how much we're motivated," Cilic said in an on-court interview as Coric celebrated with a small but vocal section of Croatian fans.
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"All weekend we played amazing tennis and Borna produced unbelievable tennis (in) the deciding rubber."
Croatia advanced to a home semi-final against France, while defending champion Britain will host Argentina in the other semi.
Britain needed to win only one of Sunday's two reverse singles against Serbia in Belgrade, and Kyle Edmund wrapped up victory for the visitors with a rubber to spare when he beat Dusan Lajovic 6-3 6-4 7-6(5).
It was the first time Britain had ever won a World Group tie without their top-ranked player as Andy Murray sat out the clash following his Wimbledon victory last Sunday.
Serbia were also without world number one Novak Djokovic.
In the other quarters, Argentina beat Italy 3-1 and France defeated the Czech Republic 3-1.
(Reporting by Andrew Both and Pritha Sarkar; Editing by Toby Davis and Larry Fine)
