France, who booked their place for the continental tournament in Poland and Ukraine by topping their qualifying group last month, lacked creativity but Remy's strike saved the day on a cold night at the Stade de France.
The hosts dominated but Jurgen Klinsmann's side could have been awarded a penalty in the 20th minute when Arsenal centre-back Laurent Koscielny, who had a night to forget, appeared to bring down Jozy Altidore in the box.
"It could have been better but some players were playing their first game for France and we made a lot of changes in the second half," coach Laurent Blanc told French TV channel TF1.
"We would like to be better but it's easy to say. It was a pleasant game but we have to be patient."
France next play Belgium at the Stade de France on Tuesday.
"It will be a more competitive game, I think," said Blanc.
FAILED TO CREATE
PSG midfielder Jeremy Menez proved a constant threat, but it was not quite enough as France failed to create clear chances.
Les Bleus had their best opportunity of the first half in the 37th minute when Franck Ribery broke down the left and set up Menez, whose swivelled attempt went over the bar.
France improved slightly after the break, coming very close when Karim Benzema's header from an Adil Rami deflection hit the bar in the 55th minute.
One minute later, Tim Howard pulled off a great save to parry away a dipping Benzema free-kick.
Blanc made a string of changes around the hour, which bore fruit in the 71st minute when Remy collected a through ball from fellow substitute Marvin Martin, muscled his way through the defence and clinically fired home.
Montpellier striker Olivier Giroud earned his first cap in the second half and made some impact although he wasted a clear-cut chance in the closing stages.
The USA lost 6-0 and 3-0 in their only two previous encounters with France in 1979.
