Italy, one of the favourites for the title, had to come from behind against the Netherlands after Botic Van De Zandschulp saved three match points in a 6-7 (6) 6-3 7-6 (7) victory over Matteo Arnaldi in a near three-hour opener.
But Sinner, who has broken into the world's top four this season and finished runner-up to Novak Djokovic at the ATP Finals on Sunday, drew Italy level by defeating Tallon Griekspoor 7-6 (3) 6-1.
And he was then brought into the deciding doubles rubber, where he and Lorenzo Sonego claimed a 6-3 6-4 victory over Griekspoor and Wesley Koolhof.
Italy will face the winner of the last quarter-final between Great Britain and Serbia in the last four on Saturday.