Mourinho, who managed Chelsea from 2004-07, returned to Chelsea in June after managing Spain's Real Madrid the past two seasons.
The victory, courtesy of goals from Oscar and Eden Hazard in the first half-hour, saw Chelsea book a meeting with AC Milan on Sunday in New York in the International Champions Cup friendly tournament.
The event, which also features Everton, Juventus, Real Madrid, Valencia and the Los Angeles Galaxy, concludes in Miami on August 7.
Chelsea had comfortably won all three of their matches in a tour of Asia before arriving Stateside, but it was Inter who had the first real chance on Thursday when Fredy Guarin unleashed a volley on goal after 10 minutes only to see it sail just wide of the post.
However, the crowd of 41,983 at Lucas Oil Stadium, home of the NFL's Indianapolis Colts, didn't have to wait long for the first goal.
Chelsea took the lead in the 12th minute as Nigeria's Victor Moses broke through to the edge of the penalty area and passed to Oscar, who curled a superb shot past Samir Handanovic and into the upper corner of the net.
Rodrigo Palacio tried to level things with a long-range shot deflected off John Terry, forcing Petr Cech to make a one-handed grab.
However, Inter would get no closer the rest of the night.
Chelsea doubled their advantage in controversial circumstances in the 28th minute. Moses was brought down on the edge of the area and a penalty was awarded despite the protests of Inter players -- whose claim he was outside the box appeared to be justified by video replays.
Nevertheless, Belgian winger Hazard calmly took his shot, sending Handanovic the wrong way to notch his second penalty in as many matches.
Inter were clearly frustrated, and Hugo Campagnaro was sent off for a nasty stamp on Terry at the hour mark.
"The red card changed (the match) completely," Mourinho said. "Not good for them because they played with 10, and not good for us, because we played against 10."
Inter will now face Valencia in the tournament's consolation bracket on Sunday.