Ireland had the better of the chances but were unable to break-down last year's finalists at the Stade Parsemain to earn an impressive point.
Captain Jayson Molumby was the first to test the Mexican goalkeeper Jose Hernandez as his 20-yard effort was well parried in the seventh minute before Adam Idah had the best chance of the game ten minutes later.
Ireland won the ball high up the pitch and the influential Connor Ronan found space on the right to find Idah in the area but after a brilliant first touch, he dragged his shot wide of the target.
Just past the half-hour, Ireland had another major chance to open the scoring. It was Ronan again who found space on the right and found the on-rushing Simon Power on the left but his shot was straight at the keeper from the edge of the area.
It was a second-half of few chances as Mexico had their only real chance of the game in the 61st minute. Dara O'Shea fouled Fernando Beltran on the edge of the area and from the resulting free-kick, Jairo Torres' strike hit the bar.
Five minutes later, Molumby had another long-range effort which went narrowly wide. It was a tense final 10 minutes where both sides had chances but the game ended goalless to secure a point for Stephen Kenny's side. If Ireland beat Bahrain in the final Group C fixture on Sunday, Ireland will be guaranteed a semi-final spot.
"We're disappointed not to win. We created a number of chances and we felt we should have taken one of them," said U-21 boss Stephen Kenny.
"Mexico were last year's finalists and they've come here with the team for the Olympics so we've got to be happy with the way we've performed and I'm really proud of the way the players have taken to the challenge of the tournament.
"We need to recover now and make sure we're at our best for the Bahrain match. The players have done brilliantly to get us into a position to qualify and we'll be ready for the Bahrain match on Sunday."
Republic of Ireland: Kelleher (GK), O'Connor, O'Shea, Masterson, Leahy, Molumby (C), Coventry, Ronan, Power (Connolly 46), Elbouzedi, Idah (Drinan 83).
Mexico: Hernandez (GK), Mozo, Govea, Angulo (C), Arteaga, Aguirre (Cordova 86), Esquivel, Beltran (Lopez 73), Torres, Yrizar (Cardona 79), Godinez (Aguirre 70).
Yellow card(s): Dara O'Shea, Jayson Molumby.