Sinner and Sonego overcame the duo of Tallon Griekspoor and Wesley Koolhof 6-3 6-4 as Italy reached back-to-back semi-finals in the premier men's team competition for the first time since three successive last-four appearances in 1996-98.
Italy, who won their only Davis Cup title in 1976, will now take on a Serbia side featuring world number one Novak Djokovic or Britain who meet in their quarter-final later on Thursday.
Botic van de Zandschulp earlier beat Matteo Arnaldi 6-7(6) 6-3 7-6(7) to give the Dutch a 1-0 lead in their bid to match their best result of reaching the semi-finals in 2001, before Sinner downed Griekspoor 7-6(3) 6-1 and levelled the tie.
Dutch captain Paul Haarhuis, who was a part of the team that lost to eventual champions France 22 years ago, made a late swap with Griekspoor replacing Jean-Julien Rojer for the doubles tie while Filippo Volandri brought Sinner in for Simone Bolelli.
The changes worked well for both teams in a high-octane clash but Sinner and Sonego won the crucial points in front of a boisterous crowd to send Italy through.
"It means a lot to me," Sinner said. "But more importantly we're through to the semi-finals. It's the first time I can experience this.
"We were under a lot of pressure. The decision to put us on the court for doubles, we have to give a lot of credit to the captain and the team, pushing us to the limit.
"It's amazing, but now we're focused on Saturday's semis."
World number four Sinner, who lost to Djokovic in the ATP Finals title clash on Sunday, got them back into the quarter-final with an impressive display against Griekspoor.
He edged the opening set before breezing through the next in only 20 minutes.
In the day's opener, Arnaldi squandered two set points and saved one as the world number 44 won a tight opening set in the tiebreak when Van de Zandschulp sent a forehand long.
The Dutchman drew level after Arnaldi dropped serve in the eighth game of the next set and the world number 51 saved three match points in the decider to prevail after nearly three hours.
