Leeds boss Daniel Farke twice led Norwich to promotion to the Premier League before being sacked in 2021.
Prioritising his fight to avoid relegation, Farke's decision to make nine changes paid off against his former club.
Chasing a first appearance in the final since 1973, Leeds went in front in the 32nd minute at Elland Road.
Sean Longstaff scored for the first time since September with a clinical finish from inside the area.
Gabriel Gudmundsson slotted home in the 43rd minute, and Joel Piroe bagged Leeds's third in the 85th minute.
Arsenal, Chelsea, Manchester City and Liverpool booked their quarter-final places on Friday and Saturday.