TULSA, Okla.—Gun owners in this state could walk down the street openly packing their pistols under legislation passed by lawmakers that would make Oklahoma the first state in seven years to allow residents to openly display their handguns in public.

Advocates of "open carry" laws have been pushing for several years to make it legal for gun owners to display their handguns in public places, arguing the practice would increase public safety and is protected under the Second Amendment.

Their...