The home side created early chances in the game, with Alessio Cerci latching on to a cross in the box, but the headed effort glazed over the crossbar.
A matter of minutes later, the Verona man was involved again as he picked the ball up on the edge of the box before firing it towards the bottom right of the net, only for the goalkeeper to make a stop.
Inter began to find their feet as the minutes ticked by and it was Mauro Icardi who came close by delivering a thumping shot from just outside of the area, but once again, the keeper prevented the ball from nestling into the back of the net.
Not long after the half-hour mark, the visitors went a goal up thanks to Borja Valero, who met Antonio Candreva's cross before hammering the ball into the bottom left corner.
After the half-time break, Perisic, who was linked with a move to Manchester United in the summer, had two chances in quick succession - one attempt flew over the bar, while the other was comfortably saved by Andrade Nicolas.
Verona pulled a goal back from the penalty spot after Samir Handanovic brought down his opponent in the box, and the keeper was punished as Giampaolo Pazzini sent him the wrong way.
The home side's comeback did not last long, though, as Perisic regained the lead for Inter by sending the ball into the top right corner with a rebound shot from the edge of the area.
Verona tried to get back into the game, but their Milan-based opponents managed to get the job done and take all three points.