Jamaica also advanced, beating Canada 3-2 in Toronto.
Mexico's César Huerta was twice allowed to retake his penalty kick in the shootout following saves by Edrick Menjívar when Salvadoran referee Iván Barton ruled the goalkeeper had moved too early.
Huerta scored on his third attempt for the 4-2 lead and Andy Najar, who spurned the US national team to play for Honduras, put his shot wide as Mexican players celebrated.
Honduras led 2-0 from Saturday's win in Tegucigalpa but Mexico tied the aggregate score in the home-and-away, total-goals series after strikes from Luis Chávez in the 43rd minute and Álvarez, who scored 1 1/2 minutes after the nine minutes of stoppage time that was signalled.
Jamaica rebounded from a 2-1 home loss in the first leg and a one-goal half-time deficit for their win in Toronto. That left the aggregate 4-4, with the Reggae Boyz advancing 3-2 on away goals.
Argentina will defend their Copa América title when the tournament is played in the US from June 20 to July 14. The draw takes place on December 7. The 10 South American nations received automatic berths.
The US, Mexico, Jamaica and Panama also advanced to the semi-finals of the CONCACAF Nations League, to be played on March 21 at Arlington, Texas. The US are likely to play Jamaica, with Panama to face Mexico.
At Mexico City's Estadio Azteca, Chávez scored from a free kick from about 30 yards that bounced past Menjívar. Mexico still needed a goal when nine minutes of second-half stoppage time was called.
César Montes lofted the ball into the penalty area and Denil Maldonado's headed clearance attempt was headed back in front by Santiago Giménez. Álvarez reacted as the ball fell at his right foot and while falling poked the ball past Menjívar for his fourth international goal.
Maldonado was given a yellow card in the 115th minute for pulling down César Huerta with a slide tackle from behind as Huerta broke in on goal.
Mexico went first in the shootout and Santiago Giménez, Johan Vásquez and Orbelin Pineda converted. Bryan Róchez saw his effort saved by Luis Malagón, while Bryan Acosta and Alberth Elis converted.
Menjivar jumped right and then left to stop Huerta, whose third attempt was outside the fingers of the goalkeeper, who leapt to the right.
In Toronto, Alphonso Davies put Canada ahead in the 25th minute for a 3-1 aggregate lead, but Shamar Nicholson scored in the 63rd and 66th minutes for Jamaica.
Ismael Koné's 69th-minute goal moved Canada back ahead, but Bobby Decordova-Reid converted a penalty kick for the Reggae Boyz's decisive goal in the 78th after Mexican referee César Ramos called a hand ball on Stephen Eustáquio.
Jamaica finished a man short after Demarai Gray was given his second yellow card in the 84th.
