The match, played as the first leg of a double-header at the New Meadowlands Stadium, finished 1-1 after 90 minutes and 30 minutes of extra-time.
Honduras converted all four penalties they attempted but two Costa Rica players, Celso Borges and Alvaro Saborio, blasted their spot-kicks against the crossbar.
Former Bristol City striker 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 Bengtson 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 equalised 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.
"It was worth the yellow card," he later told reporters.
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.
"I believe the luck factor counted," said Costa Rica coach Ricardo La Volpe.
"I think we finished (the game) much better than them. They ended up making time, lying on the floor, with a very impartial referee."
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 Mexico in the semi-finals in Houston after the defending champions beat Guatemala 2-1.
The remaining two quarter-finals will take place in Washington DC 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.
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