Folarin Balogun scored his third goal of the World Cup before being sent off with a red card in the second half, meaning he will miss the Americans' clash against Belgium.
Malik Tillman converted a free kick after that to secure the 10-man US the win on Wednesday.
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 against Tarik Muharemovic in the 64th minute.
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, Mauricio 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.
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