Spanish striker Llorente, restricted to a place on the bench for much of the season, got his first goal of the night when he slotted home from close range after fellow forward Son Heung-min burst down the right and delivered a ideal low cross.
It was a chastening experience for Tranmere, who showed spirit in the first half but swiftly ran out of ideas in the second.
Prior to this season, Tranmere Rovers have spent the last three years having to qualify for the first round proper during their time as a non-league club while tonight's opponents Tottenham Hotspur have reached the semi-finals of the competition in both of the last two campaigns.
Spurs travelled to Merseyside to take on League Two's Tranmere and came away with a club-record 7-0 victory.
With 15 minutes remaining in the match, Spurs were 6-0 up thanks to a Llorente hat-trick, a brace from Aurier and a Son goal.More news: Nadal suffers injury setback ahead of Australian Open
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After Tranmere's James Norwood lost the ball, Aurier was on hand to provide a sumptuous looping effort into the top corner of Scott Davies' net.
Llorente went close with a volley on the turn during that spell, before Tottenham went ahead in eye-catching fashion in the 40th minute.
Serge Aurier opened the scoring just before the half-time whistle with a stunning strike, before Spurs took complete control in the second half. The former Paris Saint-Germain defender's clever flick sent Son into space down the right and his low cross was slotted in by Llorente.
That 55th-minute effort was soon followed by another.
Fernando Llorente doubled their lead shortly after the restart before the Ivorian grabbed his second of the game. Naturally, the England captain promptly made it even worse for the home side.