The defending champions sailed through the group stage at the 2026 tournament but have been pushed to the limit by Cape Verde and Egypt in the knockout rounds.
The South Americans are now unbeaten in 11 World Cup games as they seek to become the first country since Brazil in 1962 to win back-to-back World Cups.
But they face a tough test against an industrious and well-drilled Switzerland team in Kansas City.
Scaloni told his pre-match press conference at Arrowhead Stadium on Friday that supporting Argentina was about "passion".
"I would like this national team to be remembered as a team that never gave up," he said.
Scaloni said he had become emotional after seeing a young fan expressing his support for the team.
"We, the technical staff, the players, we play football for this," he added. "We do not play football just for a win. And then when you see those things that come out of our heart, this is just outstanding.
"When you see a 10-year-old saying those things and talking with that passion, and everyone shouting 'Argentina', that is just so emotional. And that is the legacy that I want."
Saturday's quarter-final at Arrowhead Stadium will take Lionel Messi back to the scene of his hat-trick in Argentina's first match of the 2026 tournament, a 3-0 win over Algeria.
The Inter Miami forward is level with France's Kylian Mbappe on eight goals in the race for the Golen Boot and is now the leading scorer in World Cup history, with 21 goals across his career.
Scaloni said his star player would be the at the top for as long as he wanted to be.
"At 39 years of age, some people might think he may not rise to the challenge, but I've said before that for as long as he wants, he will be the best," he said.
"I'm not saying that because I'm coaching him, but rather because I think if he wants to, he will continue to be the best.
"Those of us who see him train, those of us that see him doing things today cannot even imagine what he was like when he was 23 years of age, when he was back in Barcelona with (Pep) Guardiola."
Scaloni said Switzerland, led by their influential captain Granit Xhaka, would be hard to beat in the last eight.
"They have a long-standing history in World Cups," said the Argentine. "They have very experienced players physically. They are strong. So it will be a tough opponent. That is for sure.
"We respect them very much, as we also do with other opponents. They eliminated Colombia. Colombia had been doing great during this World Cup."
If Argentina beat Switzerland, Argentina would face England or Norway in the semi-finals.
World Cup 2026
The 2026 World Cup will be held from June 11th to July 19th in the United States, Canada and Mexico. The tournament will feature 48 national teams and will be played in 16 modern stadiums.
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