Despite some spirited attacking play, which saw Danny Whitehead force Kiko Casilla into a couple of routine saves from long range, Mat Klich scored a spectacular third goal on a devastating counter-attack to wrap up the tie after 57 minutes.
From a cleared Salford corner, the Polish midfielder collected the ball around the halfway lane, darted half the length of the field and beat Neal from 18 yards, via a slight deflection.
“It was a great atmosphere within the stadium and our players rose to it,” said Salford manager Graham Alexander. “Quality just showed in the end.”
Salford (3-5-2) Neal; Maynard, Pond, Piergianni; Threlkeld (Gaffney, 62), Towell, Jones, Whitehead, Touray; Beesley (Lloyd-McGoldrick, 69), Dieseruvwe (Rooney, 69).
Subs (not used) Wiseman, Lethern, Jones.
Leeds (3-4-3) Casilla; White, Berardi, Davis; Shackleton, Phillips, Klich, Alioski (Harrison, 77); Helder Costa, Nketiah (Bamford, 77) Clarke (McCalmont, 70).
Subs (not used) Miazek, Dallas, Gotts, Bogusz.
Referee K Stroud (Bournemouth)