The midfielder joins the Whites from fellow Premier League side Chelsea for an undisclosed fee.
Bate, who attracted interest from a number of clubs, has penned a three-year-deal at Elland Road, running until the summer of 2024.
The 18-year-old has been part of the Chelsea youth setup for the last 10 years and captained the Blues in the 2020 FA Youth Cup final.
Last season he made the step up to Under-23s football, making 20 appearances as Chelsea finished runners-up in Premier League 2 Division 1.
He has also won caps at youth level for England at both Under-17 and Under-18 level.