On the original hardware, you can't. But through emulation, you can.
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Like most NES games, there is no way to save your progress in Super Mario Bros.
every time you complete a dark castle or spend 5 starcoins to get a mushroom you can save the game.
You have to get red flags on the level next to it.
have super skills and amazing luck. or if you're playing on an emulator, save states
In Super Mario World 2: Yoshi's Island, The game autosaves after you finish a level.