The result means Mexico will now face the United States -- 1-0 winners over Honduras earlier in the day -- in the inaugural final.
Mexico had several early chances to grab a goal, with Hirving Lozano nearly scoring off a spectacular off-balance shot in the seventh minute.
Diego Lainez also had a good look for El Tri in the 20th minute when his shot skidded inches from the goal post.
Costa Rica were able to keep buckle down defensively and stifle Mexico's potent attack from scoring.
But as full-time approached, the game turned testy with four yellow cards between the two sides in the final 10 minutes. The game was also briefly halted in stoppage time in accordance to CONCACAF's protocol on discriminatory chants from fans in the stands.
It was Mexico's third penalty-kicks win over Los Ticos following the quarterfinal of the 2019 CONCACAF Gold Cup and semifinal of the 2009 Gold Cup.