The Germany international signed a four-year deal to wrap up a transfer worth a reported €16 million (£14.5m/$18.7m), ending a season-long stay with Bayern.
Rudy's arrival will help Schalke address the loss of Leon Goretzka to Bayern after finishing second to the runaway champions in last season's Bundesliga.
"I'm convinced of the way the club wants to go and would like to be a part of it," Rudy told Schalke's website.
"And I'm really looking forward to the fans and the atmosphere at Schalke."
Schalke coach Domenico Tedesco expects Rudy to make an immediate impact in his team.
"We are happy to have Sebastian for our journey here at Schalke," Tedesco said.
"We get a very smart and technically savvy player in the squad, who has already proven his skills internationally.
"We are certain that with his qualities and experiences he will enrich our sometimes very young team and can help us directly."
Liebe Fans,— Sebastian Rudy (@BastiRudy) August 27, 2018
ich habe eine Nachricht für euch. Dabei möchte ich euch für die Unterstützung danken. Euer Basti pic.twitter.com/IrWsQ4VR7W
Rudy found himself pushed down the pecking order at Bayern following the champions' signing of Goretzka, who he will now replace at Schalke, but they recognise he made a solid contribution in his solitary campaign at the club.
"I would like to thank Sebastian for his achievements," said Bayern chief executive Karl-Heinz Rummenigge.
"He has played an important part in winning the sixth title in succession last season. We wish Sebastian all the best for his future."