A first-half header from Michael Keane put the Toffees ahead before the break, before Gylfi Sigurdsson powered home a second and Moise Kean beat Vaclav Hladky from the spot late on.
Everton, which rested new star James Rodriguez but named fellow big-money signings Allan and Abdoulaye Doucoure on the bench, hit the woodwork five times in the second half as Salford's energy dipped in the final 20 minutes of a match it had more than coped with up until that point.
Indeed the Ammies might have equalised and made for a nervous finish just before the hour mark had Richie Towell not skied a simple chance over the goal from 12 yards.
Everton will travel to Fleetwood Town for its third round clash on 24 September (AEST), but hosts West Bromwich Albion in the Premier League before that in a weekend clash.
The Baggies also progressed to the third round on Thursday (AEST), defeating Harrogate Town 3-0 at The Hawthorns.