Switzerland, which is participating in the World Cup for the sixth time in a row and the 13th in its history, took the lead through Brielle Embolo from a penalty kick in the 17th minute.
Switzerland seemed close to getting the three points and topping the group, which saw Canada draw with Bosnia and Herzegovina 1-1, on Friday, but Qatar refused to lose until the last moments.
Despite scoring the penalty kick, goalkeeper Mahmoud Abu Nada played a prominent role in keeping the Qatar national team in the match, after he blocked a number of dangerous balls, especially in the first half.
In the fifth minute of stoppage time in the second half, Khoukhi Boualem received a cross from Hammam El Amin, and headed it into the net, giving Qatar its first point in the history of its participation in the World Cup.
The Qatar national team, winner of the past two editions of the Asian Cup, lost all three matches in its only previous World Cup participation, in 2022 on home soil.