Top scorer Tom Craddock then quickly added a close-range second.
And the Hatters wrapped it up after the break with an 18-yard strike from Jake Howells and another from Simon Heslop to make it 18 goals in three games.
Salisbury's only effort of note came when Chris Shephard's first-half shot from just outside the area was parried by Luton keeper Mark Tyler.
It was Luton's seventh straight win, to help them overtake Oxford and climb into second place.
But they remain five points behind neighbours Stevenage, ahead of Saturday's now keenly-anticipated likely title decider against the leaders at Broadhall Way (3 April, 1500 BST).
Luton Town: Tyler, Asafu-Adjaye, Blackett, Pilkington, Murray, Gnakpa (Nelthorpe 80), Keane, Heslop, Howells (Newton 75), Craddock, Gallen (Hatch 77). Subs Not Used: Hall, Barnes-Homer.
Goals: Gnapka 35, Craddock 43, Howells 54, Heslop 74.
Salisbury City: Pryce, Clohessy, Giles, Martin, Ruddick, Clarke, Shephard, Adelsbury, Connolly, Tubbs (Webb 76), Flood (Reid 63). Subs: Anderson, Widdrington.
Referee: Robert Merchant (Staffordshire).