Liverpool wrapped up the tie 6-0 on aggregate but TNS, playing at Wrexham's Racecourse Ground because their own tiny ground in the village of Llansantffraid did not reach UEFA standards, made Liverpool work for their win.
The Welsh side hit a post and had a goal ruled out for a narrow offside decision after the Merseysiders missed a penalty.
Liverpool, 3-0 up from the first leg following Gerrard's hat-trick at Anfield last Wednesday, went ahead when Djibril Cisse took advantage of a slip by Chris King after 26 minutes to lash home a cross from John Arne Riise.
Substitute Gerrard then scored twice in a minute late in the game to wrap up the win.
TNS goalkeeeper Gerard Doherty saved a weak 50th minute penalty from Dietmar Hamann as the Welsh minnows gave the five-time European champions a tougher workout than they did in the first leg last week.
They also went close to a goal after 67 minutes when Steven Beck hit the post with a well-struck angled shot.
BENITEZ PRAISE
Liverpool manager Rafael Benitez was full of praise for Doherty's display over the two legs.
"The (TNS) goalkeeper saved a lot of goals," the Spaniard told ITV Sport.
"The score at the end of the tie was good. Sometimes you can score with a bit more luck and sometimes the goalkeeper makes some good saves.
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Liverpool, entering the competition at the earliest stage after finishing outside the four automatic qualifying places in the Premier League standings last season, will now face FBK Kaunas in the second qualifying round.
The Lithuanians beat HB Torshavn, the Faroe Islands champions, 4-0 in the second leg of their opening round tie for an 8-2 aggregate win.
Arturas Rimkevicius scored twice in the second half as Kaunas ran away with the tie after Andrius Velicka scored their opener a minute before halftime. Defender Irmantas Zelmikas added a late goal to seal the victory.
In the other match played on Tuesday Dinamo Tbilisi beat Levadia Tallinn of Estonia 2-0 for a 2-1 aggregate win.
The Georgians wiped out Levadia's first leg advantage when Levan Melkadze scored at the end of the first half and wrapped it up when Sergei Orbeladze made it 2-0 soon after the break.
Dinamo, who will play Brondby of Denmark in the next round, finished the match with 10 men after David Odikadze was sent off 14 minutes from time.