Some locals could be seen in the street crying as firefighters tackled the flames.
The 1893 building was formerly Battersea Town Hall. Its grand hall has hosted events, including a memorial service for those who died in the Titanic disaster in 1912.
It has been used as a community arts centre since 1974.
Last month, Labour leader Ed Miliband and deputy Harriet Harman used the venue to launch their party's arts policies ahead of the general election.
Mr Miliband tweeted: "Terrible to see photos of Battersea Arts Centre engulfed in flames. I was there in February. Thanks to the firefighters fighting the blaze."