Elvedi agreed a three-year contract with the Premier League and will again link up with Leeds manager Daniel Farke.
The 29-year-old, who previously played for Farke at Monchengladbach, said: "He is a big point for this change.
"It was just one year but I loved to have him as a manager. I learnt a lot from him and I will learn a lot now."
Elvedi is Leeds' fourth close-season signing following the arrivals of James Trafford, Harry Wilson and Tarik Muharemovic.
He played every game of Switzerland's run to the quarter-finals of the recent World Cup.
Elvedi's arrival follows the earlier exit of defender Sebastiaan Bornauw, who has signed for Hamburg on a season-long loan deal.
Elvedi could make his debut for Leeds in their Premier League opener at Nottingham Forest on Saturday.