Veteran Spanish star Sergio Ramos scored a stunning first-half header for Monterrey, delighting the decidedly pro-Mexico crowd in Southern California.
Lautaro Martinez evened it just before the break, but the Champions League finalists couldn't generate a second goal despite having nearly 62 per cent possession and only one shot on target from their opponents.
Inter coach Cristian Chivu made his debut as the Nerazzurri returned to competition at the Rose Bowl just 17 days after taking that 5-0 thrashing from PSG in Munich.
Martinez thought he scored again in the 68th minute, but it was ruled offside.
Earlier on Tuesday in Seattle, Argentina's most successful club River Plate began their campaign with a 3-1 win over Urawa Reds from Japan.
Facundo Colidio headed in a 12th-minute opener before Sebastian Driussi took advantage of Marius Hoibraten's ill-advised back-header early in the second half to nod past advancing goalkeeper Shusaku Nishikawa.
Yusuke Matsuo pulled one back from the penalty spot four minutes later only for Maximiliano Meza to score River's third header of the game.
"I am extremely happy. Very happy with the win, with the way we handled the match," Colidio said.
"We had to start out in this way, and now we keep our eyes on the next match."
In Orlando, Iqraam Rayners scored from point-blank range in the 36th minute to give South Africa's Mamelodi Sundowns a 1-0 win over Ulsan HD of South Korea after an hour-long weather delay.
And Jobe Bellingham made his Borussia Dortmund debut but could not inspire his new club as their opening match against Fluminense finished goalless in New Jersey.
The 19-year-old younger brother of Real Madrid star Jude, who joined from Sunderland last week, came off the bench in the 59th minute but was not able to swing the game in the German Bundesliga side's favour.
That meant Fluminense's 44-year-old goalkeeper Fabio, making his 1375th senior appearance, kept a clean sheet.
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