Mannone made a late mistake in the Black Cats' 2-2 draw with Manchester City to cost his side two vital points at the Etihad Stadium, leaving them six points adrift of safety with just five games left this season.
Brown told the club's official website: "I'm sure Vito will be fine - it's happened to us a lot this season - It was me last week and the quicker you get over it the better.
"There's still a chance we can do what we need to do and that's what we've got to keep hoping and striving for. The result against City can give us the confidence to go out and turn in more performances to achieve that."
Sunderland take on Chelsea at Stamford Bridge on Saturday.