Borja Valero and Perisic scored either side of Verona substitute Giampaolo Pazzini's penalty as the visitors moved back to within two points of top spot.
With Napoli, champions Juventus and Lazio all recording weekend victories, the pressure was on Inter to maintain their eye-catching start to the campaign.
And Luciano Spalletti's side were on their way when Valero, an off-season signing from Fiorentina, netted his first goal for the club towards the end of the first half, the 32-year-old profiting from Antonio Candreva's superb cross.
Verona had lost their two previous fixtures to sit 19th after 10 matches of their Serie A return, but a VAR-assisted penalty from substitute Pazzini gave them hope of a first top-flight win against the Nerazzurri in 25 years.
However, in the fashion of true Scudetto contenders, Inter promptly responded through Perisic's searing 67th-minute strike to extend their unbeaten start to 11 matches.
Verona created the first opportunity of note when Alessio Cerci's goal-bound effort was impressively blocked by Brazil defender Miranda.
The visitors soon settled and saw a well-worked opening screwed wide by in-form captain Mauro Icardi, before Matias Vecino drew a comfortable low save from goalkeeper Nicolas.
Spalletti's men had begun to assert their dominance midway through the half and, on the back of sustained possession and pressure, their 36th-minute opener came as little surprise.
Italy international Candreva produced the required class as his delightful inswinging cross from the right dropped perfectly for Valero to volley home from close range, giving Inter an assured one-goal lead at the break.
Icardi went close to continuing his own fine scoring record against Verona when his long-range attempt from Perisic's direct pass whistled over the bar.
Perisic was next guilty of scuffing an attempt into the ground, and Inter's wayward finishing came back to haunt them when the VAR awarded Verona a penalty.
Samir Handanovic, who had initially escaped punishment, was adjudged to have fouled Cerci in attempting to shovel the ball off the attacker's foot, enabling substitute Pazzini to convert the equaliser just before the hour mark.
But moments after Vecino watched a header drop on top of the woodwork, the visitors' mounting frustration was released as Perisic punished Verona for only clearing a free-kick as far as the edge of the area - emphatically thumping the loose ball through a crowd of players.
The Croatia international's stunning effort proved to be the match-winner as the title contenders comfortably closed out the victory from there, improving their league record to nine wins and two draws under Spalletti.