In every big city, there are dentists on duty during weekends and public holidays.
If you need to visit an emergency dentist, you can find one here (in German): www.zahnaerztekammer.at/patientinnen/notdienstsuche/. Pay attention to the working hours and call if you have any questions. Bring always with you the health insurance card.
Good to know
In general, doctors and dentists speak English. In case the emergency happens when a dentist isn’t available, you should go to the emergency room.