It loss meant back-to-back defeats for Chelsea who had earlier lost 3-2 to Bundesliga champions, Bayern Munich, in the ICC on Tuesday.
Thibaut Courtois saved the spot-kick, but Jovetic pounced on the rebound to give his side the lead.
But the French midfielder Kondogbia was soon trending on Twitter with many calling his unintentional long-distance lob the best own goal of all time.
New signing, Alvaro Morata, made his first start for the Blues in the match.
After Perisic's second, Inter almost made it 3-0 as he squared to substitute Joao Mario, only for the midfielder to shoot straight at Courtois.
"However, we have been playing a lot of games so close to each other, around five in a couple of weeks, and the last of them was just two days ago".
Michy Batshuayi thought he had leveled matters for Chelsea in the 88th minute, but the striker's effort was flagged offside.
Chelsea manager Antonio Conte says his team is working "very hard" ahead of next season's Premier League and Champions League.
Chelsea will face Arsenal in the Community Shield on August 6, and begin their Premier League title defence against Burnley August 12 at Stamford Bridge.
Manchester United target Ivan Perisic doubled the Italians lead with an angled drive from the left.
Goals from Stevan Jovetic and Ivan Perisic had put Inter into a commanding lead before a weird Geoffrey Kondogbia own goal gave Chelsea hope of a recovery, and Spalletti believes his players need to focus on keeping intensity levels high.
There was also shot into the side-netting, after Perisic had out-paced new Chelsea signing Antonio Rudiger, as the Inter wide man grew in confidence as the game wore on.