Sale announced a two-year deal for Webber, who will leave Bath at the end of this season.
The 29-year-old has won 16 caps and was part of England's 2015 World Cup squad, although he did not make England's elite player group announced last month by new head coach Eddie Jones.
"Rob fits perfectly into what we are trying to create at the club," Sale rugby director Steve Diamond said.
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"He is a talented all-round player who is great at the set-piece and very mobile in the loose.
"He is at the top of his game and will bring with him a wealth of experience."
Webber, who made more than 100 appearances for Wasps before joining Bath in 2012, added: " I am really looking forward to the new challenge ahead.
"I have spoken to Steve Diamond and the lads I know at the club, and they have all spoken highly about the environment at Sharks."
Diamond, meanwhile, said that Sale would be making more signings "over the coming weeks".
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