The 29-year-old has signed a three-year contract at Elland Road, with the fee reported to be around £6.8m up front and a further £1.7m in potential add-ons.
He becomes Leeds' fourth signing of the summer, following James Trafford, Harry Wilson and Tarik Muharemovic through the door, and arrives as Farke's side look to build on last season's 14th-placed Premier League finish.
Elvedi spent 11 years at Borussia-Park after joining as a teenager from hometown club FC Zürich, making 364 appearances for the German side and scoring 19 goals. Within that total, his 318 Bundesliga appearances rank eighth in the club's all-time top-flight standings, the most of any non-German player in Borussia Monchengladbach's history.
The two worked together during the 2022-23 season, when Farke coached Gladbach to a 10th-placed Bundesliga finish and Elvedi made 34 appearances in all competitions, scoring three times, including in a 3-1 opening-day win over Hoffenheim.
"I talked with the manager, I know him as well and, yeah, he is also a big point for this change," Elvedi told LUTV after completing his move. "It was just one year but I loved to have him as a manager. I learned a lot from him and I will learn a lot now."
Speaking about his first look at his new home ground, Elvedi added: "I know Leeds have a big history, Manchester United is also a big rival so we will try to beat them this season.
"It is such a big club with amazing fans and I also can't wait to play in this stadium with the fans. Everybody told me it is a special moment to go through the tunnel and see the atmosphere. I can't wait to see that."
Elvedi has earned 73 caps for Switzerland and played every minute of their run to the World Cup quarter-finals this summer, sharing the pitch there with new Leeds team-mate Noah Okafor, who joined the club from AC Milan last year. Okafor said he had spoken to Elvedi about the move during the tournament, telling him "good things about the club and the fans."
The defender's arrival effectively fills the gap left by Sebastiaan Bornauw, who departed Elland Road for a loan move to Hamburg on the same day, as Farke continues reshaping his defensive options ahead of the new Premier League campaign.
Leeds United are keen to have Elvedi available for selection as soon as his paperwork is finalised, with the new Premier League season now just days away.