World number one Novak Djokovic faces number four ranked Italian Jannik Sinner in the second singles rubber, aiming to complete the job and take his country to Sunday's final against Australia.
Musetti, chosen to play by Italy coach Filippo Volandri over Matteo Arnaldi, who played in the first singles in the last eight win over the Netherlands, edged a nail-biting first stanza.
The 21-year-old's selection was a surprise, given he had lost his last five matches, but he emerged on top after a high-calibre first set.
The Italian conceded a service break in the first game, which Kecmanovic consolidated, and the Serb forced two break points on Musetti's next serve but could not convert them.
The world number 27 battled his way back in for 5-5, the first break Kecmanovic has conceded this week, after his excellent quarter-final display against Great Britain's Jack Draper.
In the tense tie-break Musetti converted his third set point to take the lead, but Kecmanovic responded strongly in the second set.
The Serb broke at the third time of asking for a 4-2 lead, when Musetti hit the net with a backhand volley and then clinched the set with another break.
In full flow, Kecmanovic broke twice in the third to rack up a 5-0 lead, winning nine consecutive games, with Musetti getting treatment on his left hamstring as the match slipped away.
The Italian stopped the rot with a hold but Kecmanovic swept to victory with a powerful forehand winner, falling to the floor in delight.
"To come back from a set down was really tough, so I'm really proud of the way that I fought," said Kecmanovic.
"(Now) we all hope Novak is going to do what Novak does."
