Portuguese star Cristiano Ronaldo was suspended after his contested red card during the Italian side's 2-0 win over Valencia in the first round of games.
But Dybala ran riot in his first European game this season, volleying in his first following a Leonardo Bonucci long ball after five minutes.
The Argentine doubled his account on 33 minutes when he picked up a rebound from a Blaise Matuidi shot, with Juan Cuadrado setting Dybala up for his third on 69 minutes.
The Swiss played the final quarter of an hour a man down after Guinean centre-back Mohamed Ali Camara was sent off for a second yellow card.
It was the Young Boys's second defeat of the competition after they lost their opener 3-0 at home to Manchester United.