While the Mario Brothers did make an official appearance on an Atari console, that was the original Mario Bros. arcade game, not a Super Mario Bros. platformer.
So this footage you see up top, as good as it looks, isn't from a real game. It's a demo made by Atari Age forum member Sprybug, showing what Super Mario Bros. could have looked like on the Atari.
It of course looks terrible. But I love the fireball effect. It's so casual.
Super Mario Clone World 1-1 Playable Demo! [AtariAge]