This is dangerous ground for question setters.
The phrase seems to have been first used by the Winchester Repeating Arms Company, which was founded in 1866 in New Haven, Connecticut. Winchester's most famous model was the 1873, and this is the weapon with which the epithet is most commonly associated. If asked about it in a quiz, this is probably the one to go for.
However ... the Colt 'Peacemaker' pistol, which was also first manufactured in 1873 (and is perhaps better known as the Colt .45, in reference to its calibre) has also been referred to by this epithet - here, for example.
I even found a video on YouTube that covers both of these weapons, referring to them as "the Two Guns that Won the West"! (It refers to the Colt by its official title - the New Frontier Single Action Army; this one was actually manufactured in 1893, and the Winchester in the following year.)
Apparently many gunslingers in the Wild West would carry both a Winchester '73 and a Colt .45.
The waters are muddied even further by two films that were released in the 1950s. The first, from 1950, starred James Stewart and Shelley Winters; it was entitled Winchester '73. It tells the story of a prized rifle (of that model) and its "journey ... from one ill-fated owner to another and a cowboy's search for a murderous fugitive." Whether or not the featured weapon may have been "the gun that won the West" doesn't seem to be mentioned.
The second film, released in 1955, took this epithet as its title. According to what I've read on Wikipedia and IMDb, the film's plot seems to centre on "the cavalry ... depending on getting a new type of ... rifle"; but the rifle in question is not a Winchester but a Springfield 1865. Wikipedia tells us that this particular weapon was "manufactured by US Armory in Springfield, Massachusetts."
So ... if asked in a quiz, which weapon was known as "the gun that won the west" (particularly if the word "rifle" is mentioned), go for the Winchester '73. If "pistol" is mentioned, go for the Colt Peacemaker (a.k.a. the Colt .45). If asked which weapon gave the 1955 film its title, the correct answer is "the Springfield 1865". (Good luck with that ... )
If you're a question setter, and you feel compelled to ask one on this topic ... my advice would just be to be very careful how you word it!
© Haydn Thompson 2019