The 29-year-old centre-back has agreed a three-year contract at Elland Road and will reunite with manager Daniel Farke, having worked under the German during the 2022-23 season at Monchengladbach.
Elvedi highlighted his previous relationship with Farke as a key factor behind his decision to join Leeds.
"He is also a big point for this change," Elvedi said. "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."
Elvedi joined Monchengladbach in 2015 and spent 11 years with the Bundesliga club, making 364 appearances. He has also won 73 caps for Switzerland and featured in every minute of their run to the quarter-finals of the recent World Cup.
His arrival represents Leeds' fourth signing of the transfer window, following goalkeeper James Trafford, midfielder Harry Wilson and fellow defender Tarik Muharemovic.
The move also coincides with changes elsewhere in Leeds' defensive ranks, with Sebastiaan Bornauw leaving for Hamburg on a season-long loan. Striker Joel Piroe has also departed, joining West Ham on loan.
