The centre said all 200 tickets to see Brumm, Maja and Brumma were booked within 15 minutes of being made available on Thursday.
The bears are 22-month-old siblings and arrived at the centre at the end of November from Kolmarden Zoo in Sweden.
Visitors will be able to see them from 4 January.
The new arrivals join Star, the centre's resident female bear.
Comet, the centre's other European brown bear, died in August aged 30 after a short illness.
A Leisure & Culture Dundee spokesman said: "We have been absolutely blown away by the response to the preview.
"We are really excited about introducing the bears to visitors in the New Year and early indications suggest they are going to be incredibly popular."
Staff at Dundee's Camperdown Wildlife Centre said they have been "blown away" by demand for preview tickets to see its three new European brown bears.