The 'winning of the West' is a standard American phrase for the settlement of the Western parts of their country during the 19th century - cowboys and indians, gunfighters, train robbers, Pony Express, wagon trains, all that sort of stuff. There was a well-known film 'How the West Was Won' in the 1960s which is probably where the expression comes from.
It's worth pointing out, I think, that various websites interpret the lyrics differently. Some say it is "Where the West was all but won" while others give "all but one". Did you find the lyrics on a lyric site, or are they the official lyrics issued by the band?
I pronounce the words won and one differently: won rhymes with sun, one rhymes with gone. However, I know that people in many parts of the world pronounce them identically - some people to rhyme with sun and others to rhyme with gone.
Some lyrics websites are notorious for their errors. As stormwreath has indicated 'how the West was won' is a set phrase, and 'won' is probably correct. I understand '1st' and 'Amistad' refer to streets in Quemado in Texas, and Texas would be part of the American 'Old West' or 'Wild West'.