Taking to the court after American Taylor Fritz levelled the tie at 1-1 after Lorenzo Sonego gave Italy a shock lead, Bolelli and Fognini produced a flawless display in their 6-4 6-4 win to help their nation reach the semis for the first time since 2014.
Earlier, Sonego held his nerve in a tight second set to get past Frances Tiafoe 6-3 7-6(7) and put Italy - who are without the injured duo of Matteo Berrettini and Jannik Sinner - ahead.
Sonego drew first blood to go up 4-2 and saved a break point before claiming the opening set. The world number 45, who lost his previous meeting with Tiafoe in Paris this year, saved two set points in the next set to force a tiebreak.
Tiafoe then grabbed the early lead but Sonego hit back and closed out the victory on his third match point to ignite wild celebrations among Italian fans in the stands.
But Italy's joy was shortlived as Fritz dragged the 32-time winners back into the tie, battling from the brink in the first set tiebreak to beat Lorenzo Musetti 7-6(8) 6-3 and set up the doubles decider.
Italy now face the winner of Thursday's late tie between Germany and Canada for a place in the final, while Croatia will take on 28-times champions Australia in the other semi-final.
The title clash takes place on Sunday.
