City trailed 1-0 after the first leg in London and were eliminated on the away goals rule with Llorente hitting the decisive seventh goal on the night from a Christian Eriksen corner.
The hosts made the perfect start and led after just four minutes when Raheem Sterling cut inside and placed the ball beyond Hugo Lloris after being picked out by Kevin De Bruyne.
Spurs levelled matters within three minutes as the ball fell kindly to Son Heung-min on the edge of the area and his effort beat Ederson in the City goal via a deflection.
The visitors were 2-1 ahead after just 10 minutes as Son grabbed his second after being set up by Eriksen, leaving City needing three goals to turn the tie around.
Moments later the hosts were level however, as Bernardo Silva's deflected effort trickled past Lloris but City still had a mountain to climb.
It was 3-2 to City before the midway point of the first half when Sterling converted from close range from a De Bruyne assist to complete a frantic opening at the Etihad Stadium.
Spurs managed to hold the Premier League champions at bay for the rest of the first half but City improved after the break and Lloris did well to deny De Bruyne with a one-handed save as the hosts applied pressure.
Sergio Aguero gave Pep Guardiola's side the lead for the first time in the tie just before the hour mark with a first-time shot from a tight angle after being brilliantly played in by De Bruyne.
It looked as if Pep Guardiola's men were going to continue their quest for a quadruple but Spurs won a corner on 73 minutes and Llorente got the crucial touch as the ball was diverted into the goal off his hip.
There was a long break for a VAR decision but eventually the goal was allowed to stand, and City were left needing another goal.
Spurs managed to survive an onslaught until stoppage time until Sterling found the back of the net after Eriksen gave away possession cheaply, but the decision was overturned by VAR as Aguero was offside in the build-up.
Spurs are through to the Champions League semi-finals for the first time and will face Barcelona in the next round.
The two sides meet again in the Premier League on Saturday as City continue their push for domestic glory and Spurs battle for a top four finish.