First of all, sweeping is a very hard technique, which takes months and years to fully master, so don't lose your hope if you can't manage those uber clean and uber fast sweeps after a month or two.
Then, the answer to your question. You'll must mute the strings you just played with your right hand palm, or left hand fingers or whatever you find most comfortable. Starting at a very slow tempo and be patient and build up speed slowly but steady helps a lot - so don't rush into it and play over your head, because it will make you're sweeping sloppy and dirty, which will be harder to correct later, than just take your time now and get it tight from the start.
I'm no master of sweeping myself, just started out practicing on it, but as far as I know, that's what you have to do to be a truly master of sweeps some day

Good luck!