The Slovakia international was hurt just before the hour mark at St. Mary's but played on, keeping his ninth clean sheet of the season as Steve Bruce's side gained three valuable points.
He was then assessed on Monday and has now been ruled out for at least the rest of this month.
Dúbravka, 31, has not missed a Premier League match since making his debut against Manchester United in February 2018, but Karl Darlow has made five appearances in cup competitions this season.