Fistful of Dollars was a remake of the classic movie Yojimbo, which came out in the early 1960's. That was an absolute classic as a Samurai-themed movie. The Seven Samurai, also by Kurosawa inspired The Magnificent Seven... another Samurai movie turned into a Western. The movies were instantly popular.... as they broke the mold on Westerns. Which were always campi and John Wayne. Eastwood was an anti-hero at a time when anti-heros were popular. So, in short, yes they were classics pretty much from the beginning.
And Leone, who helped originate the genre, was an absolute master as a director. His ultimate movie before he died was the Gangster movie Once Upon a Time in America. An absolutely amazing epic. Leone is famous for the use of silence, few lines and 'odd sounds' like the windmill at the beginning of Once Upon a Time in the West or the ringing phone in Once Upon a Time in America. Also strange soundtracks. Also for including one huge, sweeping panaroma scene that was massively expensive at the time, but was just an amazing bit of film art. Generally, it set the scene of a town or a street or city. Always a camera pulling away to show more and more and more... Until the panorama was just breathtaking. At a time when there was no CGI, these street scenes were incredible examples of directing and set-building.
Leone is one of the great directors.... again, to answer your question... yes, classics!
Cheers,
Sirhr