Mikel Arteta earned his second win as head coach in all competitions thanks to a scrappy finish from Reiss Nelson 10 minutes after the restart.
Championship leaders Leeds had entirely controlled a first half in which Arsenal looked a shadow of the team that played so well in the narrow loss to Chelsea and 2-0 victory over Manchester United recently in the Premier League.
Arteta, though, managed to spark a second-half upturn to secure a fourth-round match away to Bournemouth.