Mike Grella pounced in the 108th and 116th minutes of Tuesday's second-round replay at Elland Road, fellow substitute Tresor Kandol struck after 109 minutes and as Kettering ran out of steam, Jermaine Beckford rubbed salt in the wounds.
Kettering player-manager Lee Harper stood between Leeds and a hatful of first half goals in normal time as he made a succession of desperate saves.
Harper twice miraculously denied Beckford and then kept out a goal-bound drive from Jonny Howson but the 38-year-old keeper was beaten by Argentine Luciano Becchio's 20th-minute header.
Robert Snodgrass cut through Kettering's defence on the left and sent over an inch-perfect cross for Becchio to head into the bottom corner in his first start since October 3.
Anthony Elding had sent Kettering's travelling fans delirious when he rose to head a 62nd-minute equaliser after Simon Heslop made the initial run and Richie Partridge supplied the cross before Leeds finished strongly in extra-time.