Tottenham Hotspur striker Son and the rest of the squad had already been confined to their Vienna hotel rooms, when not training, after six players and a staff member tested positive.
But a new round of test results, released Monday, found no additional players and only one more staffer was positive, meaning South Korea have more than the minimum requirement of 13 players available.
"Accordingly, six out of 25 national team players have tested positive for Covid-19, while 19 are available to play," a Korea Football Association statement said.
"The KFA has decided to proceed with the match, after having a discussion with the Qatar Football Association in accordance with FIFA and UEFA rules that a match can be played as long as a team has 13 Covid-free players, including at least one goalkeeper."
Kwon Chang-hoon, Lee Dong-jun, Hwang In-beom, Na Sang-ho, Kim Moon-hwan and goalkeeper Jo Hyeon-woo are the players who have tested positive, the KFA earlier confirmed to AFP.
Despite the positive cases, South Korea played Mexico on Saturday, losing 3-2, with Son -- the most prolific Asian player in Premier League history -- laying on the opening goal for Hwang Ui-jo.
Several international footballers have tested positive in the last week, including Liverpool star Mohamed Salah and Republic of Ireland midfielder Alan Browne.