Martínez stopped shootout attempts by Ángel Mena and Alan Minda, and Argentina advanced to the Copa America semifinals with a 4-2 victory on penalty kicks following a 1-1 draw Thursday night.
"I wasn't ready to go home," Martínez said. "This team deserved to advance. They complicated us a lot. They played a great match, very physical. We knew they were one of the top rivals in Copa America."
Lionel Messi, returning from a leg injury that caused him to miss Argentina's group-stage finale, put the first Albiceleste penalty kick off the crossbar, but the Albiceleste recovered and Nicolás Otamendi beat Alexander Domínguez with the winning kick.
"Our goalkeeper came up clutch," Argentina coach Lionel Scaloni said through an interpreter.
Seeking their third straight major title, Argentina advanced to the semifinals for the fifth straight Copa America and will play Venezuela or Canada on Tuesday at East Rutherford, New Jersey. Argentina are looking to win a record 16th Copa America championship.
Playing in front of a pro-Argentinian crowd of 69,456 at NRG Stadium, Lisandro Martínez gave the Albiceleste a 35th-minute lead following Messi's corner kick.
Ecuador's Enner Valencia put a penalty kick on the post in the 62nd minute after a hand ball by Rodrigo De Paul, but Kevin Rodríguez tied it in the first minute of second-half stoppage time.
With Argentina going first in the shootout, Messi tried a Panenka and hit the crossbar.
Martínez, the goalkeeper in Argentina's shootout win over France in the 2022 World Cup final, stopped Mena by diving to his left and batting the ball with his trailing hand. He denied Minda by leaping right, then followed the save with a short dance.
Julián Álvarez and Alexis Mac Allister converted to put Argentina ahead 2-0, and John Yeboah made Ecuador's third attempt.
Gonzalo Montiel gave Argentina a 3-1 lead, Jordy Caicedo converted to keep Ecuador alive and Otamendi got the winner, tucking the ball inside the post just past the outstretched arm of Domínguez.
Despite the victory, Scaloni said he "did not enjoy the win."
"We have reached the semis, and given the great level of performance of every national team, I think it's a great achievement," he said.
Lisandro Martínez had put Argentina ahead in the 35th minute with a header after Mac Allister headed on Messi's corner kick. Rodríguez scored when he headed in John Yeboah's cross.
"The quarterfinals had us playing against the most difficult opponent, and I thought we delivered well," Ecuador coach Félix Sánchez Bas said through an interpreter. "We showed this team can compete against any opponent."
Under Copa America regulations, the match went straight from the end of regulation to a shootout, without extra time.
Messi, who sustained the injury on June 25 against Chile, got a loud ovation when he entered the field 45 minutes before kickoff. He remains without a goal in three Copa games this year.
"I think he finished the match well," Scaloni said. "We asked him how he was feeling for the last time four minutes before the end."
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