The measure now goes back to the State House, where the Senate killed a similar measure last year, leading to a large tax increase for homeowners.

After four hours of tense debate and a series of failed amendments, the Boston City Council overwhelmingly approved a mayoral home rule petition that seeks to shift more of the city’s tax burden onto businesses to provide relief to homeowners.

Boston City Council has approved a change to the city’s tax structure for the third time, putting it in the hands of Beacon Hill legislators to provide relief for homeowners struggling under high tax bills.