Questioned for an unconvincing debut season at Stamford Bridge, the reported AC Milan target turned in a promising 65 minutes as the Blues prevailed from the spot after a 1-1 draw in normal time.
Argentina's Willy Caballero also enjoyed a measure of redemption in Nice as the goalkeeper put a forgettable FIFA World Cup behind him by denying Milan Skriniar and giving Chelsea a 5-4 shootout success for its second straight win under new boss Maurizio Sarri.
Morata was particularly influential early on, going it alone in the eighth minute with a superb turn and shot that fell kindly for Pedro to tuck away the opener, just as he did in the 1-0 defeat of Perth Glory that kicked-started the Sarri era.
The early strike would not be enough on this occasion, however, as Inter stemmed the tide and equalised soon after half-time through Roberto Gagliardini.
Both sides failed to put away late opportunities to snatch victory in normal time and Chelsea ultimately held its nerve, keeping its winning run intact ahead of a meeting with Arsenal in Dublin on Thursday (AEST).