The title holder opened the scoring with a header from defender Lisandro Martinez (35), then defender Kevin Rodriguez equalized for Ecuador with a header in the last moments (90+1).
Star Lionel Messi missed the first penalty kick, before his goalkeeper Emiliano Martinez saved him by blocking two kicks.
Argentina, the record 15-time champion, will meet the winner of Venezuela-Canada in the semifinals on Tuesday in East Rutherford, New Jersey.
Messi, the eight-time world player of the year, started after being doubtful for the match due to pain in his right adductor muscle.
This time, coach Lionel Scaloni started the tournament’s top scorer Lautaro Martinez (4 goals) alongside Messi, before he was substituted in the 65th minute in favour of youngster Julian Alvarez.
Argentina, the 2022 World Cup champion in Qatar and one of the favourites to win the title this year, topped its group in the first round, after beating Canada 2-0, Chile 1-0 and Peru 2-0 with a reserve line-up.
Argentina has never lost to Ecuador since 2015 in World Cup qualifiers, and has never lost to them in their history of encounters in the Copa America.
In contrast, fourth place is Ecuador’s best ever appearance, in 1959 and 1993. After losing the opening match to Venezuela 1-2, Ecuador qualified by beating Jamaica 3-1 and drawing 0-0 with Mexico.
In the remaining quarter-finals, Brazil, who failed to top their group, will play a highly anticipated match against Uruguay, the record 15-time champion, on Saturday in Las Vegas, Nevada, while Colombia will meet Panama in Glendale, Arizona.
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