The match had threatened to be overshadowed by fresh controversy involving the Video Assistant Referee, who was repeatedly called upon in a first half that ended level at 1-1.
But Llorente’s treble put the result beyond doubt with Son Heung-Min, who had opened the scoring on the night, making it 5-1 in the 65th minute.
Spurs, eight-times FA Cup winners, will now be away to Premier League rivals Swansea — Llorente’s former club — in the quarter-finals as they bid to win a first major trophy since lifting the 2008 English League Cup.
“I feel good,” Llorente told BT Sport. “I’m happy because the team have won and it’s important to be in the quarter-final. We know Swansea very well — they’ve improved a lot and it will be difficult.”
– ‘Too early to judge’ –
Tottenham manager Mauricio Pochettino, reflecting on the use of the VAR, said: “It’s too early to judge the system but it’s not easy. I think the referees in the Premier League are the best in Europe”.
Rochdale boss Keith Hill was proud of his players despite an ultimately large loss.
“I’m not bothered about the scoreline,” Hill insisted. “I wanted the lads to show bravery.”
Third-tier strugglers Rochdale were unchanged from the side that held Tottenham to a 2-2 draw in the original tie while Pochettino made nine changes from the team which snatched a 1-0 league win at Crystal Palace last time out.
Spurs thought they had taken an early lead thanks to Erik Lamela’s close-range finish.
But referee Paul Tierney, after consulting the VAR, ruled it out for what appeared to be a foul by Spanish striker Llorente on Harrison McGahey in the build-up.
Tierney consulted the VAR, being trialled in English cup competitions this season, again to deny Lucas Moura a penalty but Spurs did take the lead soon afterwards when Son slotted home in the 23rd minute.
With Harry Kane on the bench, Son was Spurs’ designated penalty-taker.
He got a chance from the spot when, after another lengthy VAR consultation, Tottenham were awarded a penalty for a foul by Matt Done on Kieran Trippier.
The South Korea star then put the penalty into the back of the net but Tierney disallowed the ‘goal’ because Son had stopped in his run-up.
Pochettino’s furious mood would not have been improved when Rochdale equalised in the 31st minute, with Stephen Humphrys latching on to a fine ball over the top from Andy Cannon before shooting past Tottenham goalkeeper Michel Vorm.
Cannon, unluckily, hit the post before Llorente ended any thoughts of a huge upset early in the second half.
He made it 2-1 in the 47th minute when he chipped over goalkeeper Josh Lillis from Moura’s through ball.
Six minutes later, Llorente gave Spurs a two-goal lead when, from Moura’s cross, he finished from close range.
Just before the hour Llorente completed his hat-trick after heading in from six yards out following a cross by the influential Son.
And it was Son himself who made it 5-1 when, after Moura had released Lamela down the left, he tapped into an empty net from the edge of the six-yard box.
Now only an abandonment could deny Spurs but Tierney was content merely to ask the groundsmen to clear the lines of snow.
And deep into stoppage time, with virtually the last kick of the game, substitute Kyle Walker-Peters scored his first Spurs goal to make it 6-1.
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