For the second game running, Spain international Pedro struck after just eight minutes in Nice, slotting in a rebound after Inter keeper Samir Handanovic saved an Alvaro Morata effort.
Inter however levelled early in the second half through Roberto Gagliardini, whose drilled shot proved too much for Argentina’s Caballero after fellow halftime substitute Tiemoue Bakayoko gave the ball away.
New Chelsea signing Jorginho nearly won it with a curling effort that forced Handanovic into a fine sprawling stop, while Tammy Abraham couldn’t hit the target as he tried to steer home an Emerson cross.
At the other end, twenty-year-old Argentine Lautaro Martinez, who joined Inter from Racing Club at the start of July, almost snatched a winner for the Italians in stoppage time when his sharp first-time strike was well kept out by Caballero.
Caballero saved from Skriniar with his legs in the shootout before Cesar Azpilicueta converted the decisive spot-kick to see Maurizio Sarri’s men through.
The Blues next take on Premier League rivals Arsenal in Dublin on Wednesday before wrapping up their pre-season campaign at home to Lyon on August 7.