Sylvain Ripoll’s France is looking for a first Olympic medal since gold in 1984, but collapsed in the second half at the Tokyo stadium.
“We are obviously disappointed with the result but also with the balance of our game,” said the former boss of Lorient Ripoll.
“To compete we would have needed a big performance, but it wasn’t.
Mexico took the lead less than two minutes after the break thanks to Alexis Vega, before Sebastian Cordova doubled their advantage in the 55th minute.
35-year-old captain André-Pierre Gignac, whose last game for the senior France team was the Euro 2016 final loss to Portugal, retired from the penalty spot 21 minutes from time.
But any idea of a comeback has been dashed as Uriel Antuna and Eduardo Aguirre scored late for the 2012 champions Mexico.
France now faces a difficult task to get out of Group A with a place in the top two, ahead of the matches against South Africa, affected by Covid, on Sunday and Japan on July 28.
“The first thing we need to do now is to digest this disappointment,” Ripoll added. “We only have two days. There’s no time to lose.
Japan eliminated South Africa, two of whose players tested positive for Covid-19 on Sunday, 1-0 thanks to Takefusa Kubo’s strike in the second half.
Brazil, who first won gold when inspired by Neymar at home five years ago in Rio, beat Germany 4-2 in a dramatic game which saw striker Richarlison achieve a hat-trick in the first half.
Everton’s man scored three times in the first half hour, but Matheus Cunha missed a penalty and Germany then threatened an unlikely comeback in the second half.
Bayer Leverkusen’s Nadiem Amiri closed the gap before the hour mark, and although Maximilian Arnold was sent off shortly after, substitute Ragnar Ache reduced the deficit to one goal within seven minutes of the remaining 90.
But it was another Leverkusen player, Paulinho, who came off the bench and put the game to bed for Brazil in the fourth minute of stoppage time.
Ivory Coast also collected three points in Group D as Franck Kessie of AC Milan scored the decisive goal in a 2-1 win over Saudi Arabia.
Spain held by Egypt
Spain struggled in Sapporo as a six-man squad that reached the Euro 2020 semi-finals dominated possession but created few chances in a 0-0 draw against the ‘Egypt.
They came closest to the goal when Real Madrid midfielder Dani Ceballos, who recently spent two seasons on loan at Arsenal, crushed a first-half shot against the post.
Spain, home winners in 1992, will next face Australia in three days before their final Group C game against Argentina.
Australia quickly took control of the table, as goals from Lachlan Wales and Marco Tilio saw them beat Argentina to 10 2-0.
New Zealand got off to a successful start in Group B as Burnley striker Chris Wood, one of the team’s three major players, claimed a 1-0 victory over South Korea in Kashima with his winner in the 72nd minute.
It was their first-ever victory in Olympic football, having managed only one draw in each of the 2008 and 2012 tournaments.
An own goal saw Romania beat Honduras 1-0 in the group’s other game.