Co-hosts the United States progressed to the next round on Wednesday with a 2-0 win over Bosnia-Herzegovina, but had to play with 10-men from the 64th minute after Balogun was red carded for his tackle on Tarik Muharemovic.
Balogun had earlier opened the scoring for his third goal of the tournament, but was genuinely surprised when he was shown red following a VAR intervention.
"It never was intentional," Pochettino said. "It's never a red card. Never.
" ... If the intention is to damage the opponent, OK, I understand. But that never was.
"It was a normal action in football that you are fighting for the ball and your feet land."
The team also stood by Balogun, who became only the third player to score and be sent off in a World Cup knockout game after Brazil's Ronaldinho in a 2002 quarter-final against England and France's Zinedine Zidane in the 2006 final against Italy.
Balogun, who matched Landon Donovan in 2010 for the second-most goals by an American in a World Cup behind only Bert Patenaude's four in the initial tournament in 1930, received the Americans' fifth World Cup red card after Eric Wynalda against Czechoslovakia in 1990, Fernando Clavijo against Brazil in 1994, and Pablo Mastroeni and Eddie Pope against Italy in 2006.
"We told him we've got his back. We're a team of 26, not just one," defender Chris Richards said. "One man's down, the next guy steps up."
Though down to 10-men, Malik Tillman converted a free kick late to secure the win.
Balogun dominated the first half with his goal and several other chances that helped the US control the match early, but then they had to scramble a man down after his foul on Muharemovic.
Tillman sealed the win when he scored from the free kick from just outside the box in the 82nd minute with goalkeeper Nikola Vasilja getting a hand to the shot but unable to keep it out.
It gave the US just their second World Cup knockout-round win.
The Americans made it to the semi-finals in the first tournament in 1930 by winning their group, and a round-of-16 matchup against Mexico in 2002.
The win over Bosnia in the round of 32 in this year's expanded tournament sets up a matchup on Monday in Seattle against Belgium, who came from two goals down to beat Senegal 3-2.
As expected, Pochettino's men were bolstered by their army of fans who chanted "U-S-A! U-S-A!" in the closing minutes.
The win snapped a 10-game losing streak for the US against European teams dating to a tie against England in the 2022 World Cup.
It marked the first win in the World Cup for the Americans against a European team since a win over Portugal in the 2002 opener.
They were winless in 13 straight World Cup matchups against European teams since then, including a 2-1 loss in extra time to Belgium in the round of 16 in 2014.
