Much of the damage was inflicted by Anthony Limbrick’s side during the first half of the Uefa Europa Conference League second qualifying round clash.
Leo Smith gave Saints an early lead before defender Danny Davies netted twice and Declan McManus also struck to make it 4-0 at the break. Jordan Williams added a late fifth to round off a wonderful night for in-form Saints.
The return leg will be played at Park Hall next Thursday, July 29.
Saints opened up an eighth-minute lead at the Hikvision Arena.
McManus, having been initially found by Smith in the centre circle, ran at the home defence and chose the perfect time to pick out Smith, with the midfielder taking a touch before lifting his shot past goalkeeper Deividas Mikelionis.
It was Smith’s fifth goal in as many European appearances for TNS.
Saints goalkeeper Paul Harrison had to be alert to keep out a low effort from Linas Pilibaitis, but the visitors then helped themselves to three goals in quick succession leading up to half-time.
Defender Davies, signed from Connah’s Quay Nomads last month, doubled the advantage in the 34th minute, directing home a header from skipper Chris Marriott’s inviting free kick from the right.
Davies quickly struck again, heading home after Marriott had nodded the ball back into the penalty area when the home side were unable to clear a Smith corner.
It got even better for Saints five minutes before the break when Davies this time turned provider as his low cross from the right was confidently swept home by McManus.
Egidijus Vaitkunas struck a post with a long-range effort for Kauno Zalgiris just before half-time.
Saints, having knocked out Glentoran from Northern Ireland in the previous round, continued to keep it tight defensively in the second half. They increased their winning margin three minutes from time when substitute Williams drove home a low shot off a post from 20 yards to complete a stunning result.
TNS: Harrison, D. Davies, K. Davies, Astles, Marriott, Clark (Cornish), Routledge, Smith, Robles (Williams), McManus (Ebbe), Cieslewicz (Rees). Subs not used: Redmond, Baker, Bradford, Roberts.