Schools and colleges will remain shut for a week and cinemas for three days. The city, India's commercial capital, is in Maharashtra state, which has seen 11 of India's 19 swine flu deaths - three in Mumbai alone.
Maharashtra Chief Minister Ashok Chavan told a television channel that the temporary closure was "only a prevention measure".
"There is no panic...we are going according to the wishes of the people and that is why we are shutting down for about seven days," he told CNN-IBN.
Three of the seven days are holidays, including the weekend, so the shutdown is effectively for less than a week, Mr Chavan said. A number of Bollywood film releases on Friday have been delayed after cinemas in the capital of the country's booming film industry shut their doors.
More about H1N1 virus in Mumbai click on BBC online.