After a goalless 90 minutes, Mexico secured victory on penalties as captain Jesus Angulo stepped up for the fifth and final penalty to seal a 4-3 win.
A much-changed Ireland side dominated the vast majority of the game and went closest through Canice Carroll in the first-half. With 10 minutes gone, Aaron Drinan found the onrushing Carroll just outside the area but he dragged his shot wide.
Just before the break, Carroll went close again as Lee O'Connor's right-sided free-kick found the midfielder and his header was deflected wide.
Ireland continued to create chances in the second-half and had two chances in the final five minutes of the game. Substitute Connor Ronan crossed for fellow substitute Aaron Connolly and he found the impressive Jack Taylor but he placed his effort wide from the edge of the area.
In added-time, Taylor had another opportunity to secure third spot. Substitute Darragh Leahy's long-throw was flicked on to Taylor but his half-volley went over the bar from six yards out.
Connor Ronan saw Ireland's first penalty saved but substitute goalkeeper Conor Kearns pulled off an excellent save to level the penalty shoot-out. Leahy was forced to repeat his penalty by the referee and the keeper saved the second penalty for captain Angulo to win the shoot-out.
Republic of Ireland: Kelleher (GK (Kearns 65), O'Connor, Masterson (C), Wilson (Leahy 58), Scales, Carroll, Lennon, Taylor, Knight (Ronan 75), Mallon (Connolly 58), Drinan (Idah 77).
Mexico: Jurado (GK), Mozo, Angulo (C), Calderon, Vazquez, Esquival (Aguirre 75) Beltran (Aguirre 60), Medina, Cardona (Torres 84), Ruiz, Cisneros (Lopez 60).
Referee: M. Gonzalez Alveal (CHI)
Yellow Card(s): Carroll