(Liverpool win 6-0 on aggregate)
Steven Gerrard scored twice inside a minute late on as Liverpool cruised comfortably into the Champions League second qualifying round.
Djibril Cisse gave Liverpool a first-half lead, and Didi Hamann had a penalty saved by Ged Doherty.
Gerrard then emerged as a substitute to deliver a double strike in the last six minutes to rubber-stamp a tie against FC Kaunas of Lithuania.
TNS' best chance came when Steven Beck saw a powerful drive strike a post.
Liverpool boss Rafael Benitez fielded a strong side, despite a comfortable advantage, although he chose to leave first-leg hat-trick hero Gerrard on the bench, along with Fernando Morientes.
And with Southampton's Peter Crouch waiting in the wings to seal a £7m move to Liverpool, Cisse took the opportunity to press his claims.
He had already been denied when clean through on Doherty, but made no mistake when he put Liverpool ahead after 26 minutes.
Chris King missed John Arne Riise's cross, and Cisse controlled before flashing home a low finish.
Liverpool dominated possession totally, but TNS were resilient and managed to restrict the Premiership side to occasional threats.
Phil Baker needed to smuggle an Anthony Le Tallec effort off the line after 33 minutes, while Boudewijn Zenden's cross flew inches wide in first-half stoppage time.
Liverpool should have doubled their lead after 48 minutes when they were awarded a penalty after Jamie Wood fouled Darren Potter.
But Hamann's penalty was weak and Doherty was able to save comfortably diving to his left.
Doherty was in action again as Liverpool's domination increased, blocking a free-kick from Steve Finnan.
TNS' moment of glory almost arrived after 66 minutes when Beck was desperately unlucky to see a curling drive hit the post with Pepe Reina helpless.
It was the signal for Liverpool to send on captain Gerrard for the impressive Xabi Alonso.
TNS keeper Doherty was having an outstanding game, and he performed heroics again to block a powerful drive from Liverpool substitute Luis Garcia.
But even Doherty was powerless once Gerrard exerted his influence, firing a long-range drive into the bottom corner after 84 minutes, then adding another less than a minute later with a shot the goalkeeper should have saved.
TNS: Doherty, Naylor, King, Baker, Evans, Holmes, Ruscoe, Beck, Toner, Hogan, Wood. Subs: Acton, Lloyd-Williams, Wilde, Lawless, Leah, Ward, Jackson.
Liverpool: Reina, Finnan, Hyypia, Riise, Cisse, Le Tallec, Alonso, Hamann, Carragher, Zenden, Potter. Subs: Carson, Gerrard, Luis Garcia, Josemi, Warnock, Whitbread, Morientes.
Referee: Carlo Bertolini (Switzerland)