The agreement with the Ohio Automobile Dealers Association was negotiated on Tuesday and approved by a Senate panel.
The bill allows Tesla to keep its existing stores in the Columbus and Cincinnati areas, and allows a third to open in Cleveland, but prohibits the company from opening any other stores.
The bill allows no other automaker to operate manufacturer-owned stores in the state.
Ohio was among states proposing to block Tesla from setting up additional direct-sales galleries on grounds they undercut traditional auto dealerships. Dealers saw Tesla as a threat to a system in which almost all dealerships are independently owned franchises that are separate from manufacturers.