The match, played as the first leg of a double-header at the New Meadowlands Stadium, finished 1-1 after normal time and 30 minutes of extra time.
Honduras converted all four penalties they attempted but two Costa Rica players, Celso Borges and Alvaro Saborio, missed theirs, both blasting their spot-kicks against the crossbar.
Saborio also failed to convert a penalty in the second half of regular time when Honduras goalkeeper Noel Valladares, named man of the match, guessed correctly and dived to his right.
After a scoreless first half, Honduras opened the scoring in the 50th minute when striker Jerry Bengston tapped in his third goal of the tournament.
Javier Portillo broke free in the Costa Rica box and fired a low shot that goalkeeper Keylor Navas deflected to the unmarked Bengston.
Costa Rica equalized five minutes later when Dennis Marshall headed in a corner from Cristian Bolanos then won a disputed penalty when Marco Urena fell in the box after defender Victor Bernandez tugged his shirt.
Valladares was given a yellow card after protesting that Saborio had not placed the ball on the spot but regained his composure in time to pull off the save.
Both sides had chances to win the game in extra time but were unable to find the mark, resulting in the penalty shootout, which was sealed by Bengtson.
Honduras have now reached the semi-finals in three of the last four editions of the tournament for North and Central America and Caribbean teams, while Costa Rica bow out on penalties for the second time in succession.
Honduras will play the winner of Saturday’s second quarter-final, between Guatemala and defending champions Mexico, in the semi-finals in Houston.
The remaining two quarter-finals will take place in Washington DC on Sunday with the United States playing Jamaica and Panama facing El Salvador.
The eventual winner of the tournament will represent CONCACAF at the 2013 Confederations Cup in Brazil.
