The Scots made an impressive opening to the match building a 14-0 lead and then a 19-6 advantage through tries by Rory Darge, Huw Jones and Ben White, but Italy hit back.
Tommaso Allan kicked two penalties before another brace of three-pointers to reduce Scotland's lead to seven points, and when Juan Ignacio Brex intercepted Finn Russell's pass to level things up, the majority of Murrayfield was concerened a nightmare Six nations start could happen.
Smart work from Darcy Graham sparked Scotland into life when he received the ball on the right-hand wing, cut inside and sped away before Jones was on hand to collect and score.
Graham was also involved in their fifth score as he took the ball off the shoulder of Russell to take them deep into Italian territory. The ball eventually went from left to right where Jones wriggled through three defenders to cross the whitewash for his hat-trick.
The tries gave Scotland enough of a buffer to see out the rest of the game, despite a late pressure from Italy which saw them create a number of chances.
Speaking after the game, Player of the match Huw Jones told BBC Sport: "I'm pretty happy.
"It's not every day you manage to get on the end of three so thanks to the boys for setting me up.
"It was a big turning point [Jones' second try], there wasn't really anything on for him but Darcy [Graham] will find a gap when there isn't one. I just had to try and keep up with him.
"I don't want to say that our confidence is sky high, we're happy with the win today but our focus is on taking it week by week."