The countries' top players face each other in Sunday's first reverse singles match with Murray looking to seal the victory for Britain against Fognini.
"We managed to convert some chances in the fourth set, Colin came up with some good returns in the end and I felt like we deserved to win," Murray said. "Bolelli played an excellent match, he made it really tough for us and was hitting the ball huge from the back of the court.
"It should be a tough match (against Fognini). Fabio has played very well this year so it'll be a great challenge. But I hope I can have a good match and get us to the semis."
Italy has not lost at home to Britain since 1926 and is hoping to reach the semifinals for the first time in 16 years.
Murray gave Britain the first break of the match with a backhand volley winner past Bolelli. Both Italians then argued with the referee in the final game of the first set following what appeared to be a double-hit from Fleming on a point won by the British team.
That appeared to affect the Italian mindset and Britain raced into a two-set lead, sealing it with a cross-court smash from Fleming as the duo held to love.
The match then turned on its head at the start of the third set.