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Croatian Perisic opened the San Siro goal blitz after 23 minutes and added two more in the second half, while Inter captain Mauro Icardi made it 16 for the season seven minutes before the break and defender Milan Skriniar found the net on the hour mark.
Luciano Spalletti's Inter are now the league's only unbeaten team after 15 games, after Napoli lost 1-0 at home to champions Juventus on Friday.
Inter are top of Serie A for the first time since January 2016, one point ahead of Napoli and two above Juventus, who are chasing their seventh straight league title and host Inter on Saturday.
"It's not important who scores the goals, only the team counts, we've waited a long time for first place, now we have to stay, we're going to Turin to win against Juve," said Perisic.
"It's not the title match but we want to try to bring home three points like in all the matches," added Spalletti. "But we have shown we have character and players of their level so it will certainly be a great challenge."
Despite the absence of key players Roberto Gagliardini, Matias Vecino and Joao Miranda, Inter proved deadly efficient against an outclassed Chievo.
Icardi moved top of the goalscoring charts but it was Perisic who stole the show on a night where there were double celebrations for Inter fans as AC Milan let points slip against rock-bottom Benevento.
Keeper scores for Benevento
Goalkeeper Brignoli snatched newcomers Benevento's first ever Serie A point deep into stoppage time to spoil Gattuso's debut and spark wild celebrations for the hosts and their fans after a thrilling 2-2 draw.
With the home side a goal down and pushing ten-man Milan back in search of an equaliser, Brignoli, on loan from Juventus, headed home Danilo Cataldi's free-kick five minutes into injury time to end the southern side's record losing run.
Benevento had surpassed Manchester United's 12-game losing start, a record for the top five European leagues established 87 years ago, on November 19, and were stuck on zero points after 14 games coming into the match with Milan.
"Someone from the bench told me to go up, there was nothing to lose, I went to jump and I closed my eyes," said Brignoli.
"It's a very strong, indescribable feeling. I dedicate the goal to all the people who are living a dream that they hoped would be better.
"We have lost so many games undeservedly at the end, even with Juventus, for once after three months of sacrifice by everyone it is nice to be able to celebrate."
Giacomo Bonaventura had opened the scoring for Milan after 38 minutes at the Stadio Vigorito, before George Puscas pulled Benevento level five minutes after the break.
Croatian forward Nikola Kalinic got Milan back in front seven minutes later only for Alessio Romagnoli to be sent off for a second bookable offence, leaving the visitors a man down for the final 15 minutes.
But in the final seconds Brignoli jumped highest to meet a desperate last free-kick.
"It would have been less painful to suffer a stab wound than that goal," said Gattuso, who took over in midweek after Vincenzo Montella was sacked as the seven-time European champions sit eighth, way below the Champions League ambitions of their new Chinese owners.
Elsewhere Lazio rallied to a 2-1 win at Sampdoria to move two points behind fourth-placed city rivals Roma.
Colombian striker Duvan Zapata put the hosts ahead after 56 minutes before Serbian Sergej Milinkovic-Savic equalised ten minutes from time, and set up Ecuadorian Felipe Caicedo for an injury-time winner.
Fiorentina handed new Sassuolo coach Giuseppe Iachini a first defeat, 3-0, to move ahead of Milan into seventh with Bologna held 1-1 at home by Cagliari.
