WASHINGTON—The House on Tuesday overwhelmingly approved new chemical safety rules designed to overhaul federal regulation covering thousands of chemicals in daily use, a rare bipartisan action in a year when Congress is torn by presidential politics.

Trade groups representing Dow Chemical Co., DuPont Co. and others pushed for the legislation as states, and even large retailers like Wal-Mart Stores Inc., Lowe’s Cos., and Target Corp., were stepping in with their own, sometimes far-reaching rules over concerns about chemical...