More than 10,000 people passed through the doors to the SeaCity Museum in the first 12 days since its opening.
The figures include 1,000 city residents who took up a council offer to visit the Havelock Road attraction for free.
The museum sold out on its first full day, with an estimated 1,600 visitors.
The city’s leisure and culture boss, Councillor John Hannides, said the visitor numbers were “exceptional” and “exceeded expectations”.
“It just shows the reach of the attraction right across the country. The feedback we’ve had has been overwhelmingly positive,” he said.
The museum opened on April 10 to mark the 100th anniversary of Titanic setting sail from Southampton on her ill-fated maiden voyage to New York.
The council had forecast that the museum would attract more than 150,000 visitors a year, but it could smash that figure if current interest holds up.
Civic leaders say they will not be making a great loss on the offer of free tickets to 100,000 city households as they only expected about 12,000 city residents to have paid to visit – worth £70,000 to £80,000 in income.
SeaCity tells the story of Southampton’s crew on Titanic, and the impact the tragedy had on the city.
The museum also features a major maritime-led exhibition which tells the stories of people who have arrived and departed in the city since prehistoric times.