The Canadian team won their first Davis Cup championship on Sunday by defeating Australia 2-0 under the leadership of Felix Auger-Aliassime and Denis Shapovalov.
In the semi-finals, the men in blue led by captain Filippo Volandri will face the winner of Germany's international match against Canada.
Croatia's world number 17 Cilic, a former US Open champion, rallied to defeat 13th-ranked Pablo Carreno Busta 5-7, 6-3, 7-6 (7/5).
Eight teams remain and will compete in the quarterfinals beginning on Tuesday. In the first quarterfinal, Australia will face the Netherlands.
"The medical experts have advised that I am still not able to compete at my highest level following the foot injury l suffered in Naples," he said.
Alcaraz was beaten by Auger-Aliassime in what was a belter of a performance by the Canadian in Spain's previous Davis Cup match.
The shock of the night during Holland vs Great Britain came in the second singles match where Botic van de Zandshculp got the better of Cameron Norrie 6-4, 6-2.
Jean-Lennard Struff was in action in the first match against Zizou Bergs of Belgium and after two sets, the German recorded a 6-4, 7-6 (11-9) victory.
Despite having a fairly strong team on paper, Argentina got off to a disastrous start as they lost 2-1 to Sweden in the first round.
Cameron Norrie was the only one who scored a win for the Brits against USA as he defeated Taylor Fritz 2-6, 7-6 (7-2), 7-5.