City won the thrilling encounter 4-3, but Llorente's goal 17 minutes from time saw Spurs progress on away goals after a 4-4 aggregate draw.
Raheem Sterling opened the scoring, but City was swiftly pegged back by a Son Heung-min brace, meaning it needed to net another three times without reply.
City did manage to score another three goals, with Bernardo Silva, Sterling and Sergio Aguero converting to seemingly send them through.
But VAR drama late on saw Llorente's goal stand despite the ball appearing to hit his arm, while Sterling had a stoppage-time effort disallowed after Aguero had strayed offside.
Although that brought a cruel end to City's celebrations, Guardiola remains a backer of VAR.
"I support VAR, it is just to see that Llorente's goal maybe from one angle is handball, from the referee's it is not," Guardiola said. "The referee must be helped, if it's offside, it's offside.
"I am fine for the fair football and the fair decision. In the future and even the present it will be fair and that's all."
Guardiola was less forthcoming about the decision to disallow Sterling's goal, which would have sent City through.
"I don't know. I saw the action, and the referee and the ones from above decide it is offside," he said.
"It's tough. In that moment you say; 'It's there, what we've done all game.' We missed a penalty in the first leg, we created an incredible amount of chances. We reacted incredibly well [to the early goals].
"I am incredibly proud of the fans. It was an incredible night for us. I have never seen the stadium like this. Thank you so much for the reaction, the fans know we gave absolutely everything."
Spurs will play Ajax in the semi-finals.