With striker Kane sidelined for several weeks due to ankle ligament damage, Mauricio Pochettino's side received an inspirational performance from Eriksen at the Liberty Stadium.
"The team played so well, 3-0 - we deserved more".
It took just 11 minutes for Christian Eriksen to open the scoring, on the back of some good Spurs pressure.
"For a team like us to win this game, two things must happen: we must be at our best level and Tottenham must have a bad day".
"It's a nightmare. I feel so sorry for the people trying to use that system", said Pochettino, who had also seen VAR dominate proceedings in the fifth-round replay against Rochdale.
"It keeps a very good momentum for us".
"Michel's save at the start of the second half was massive for us but it was the only chance they created. It's important to go to the semi-final and to have the possibility to fight for the trophy", Pochettino gushed after the tie at the Liberty Stadium on Saturday.
Tottenham then dominated much of the game after going ahead as Eriksen's flawless ball over the top found Heung-Min Son and he finished magnificently off the crossbar but the South Korean star was flagged offside and VAR decided that was the correct call.
Swans manager Carlos Carvalhal has spoken before of making the opposition "listen to our music" but Eriksen was the conductor while Lamela, Son Heung-min and Lucas Moura played a sweet symphony, their interchange and movement beguiling Swansea.
Once again, Swansea were chasing shadows and when Sissoko played in Lamela the Argentinian skipped past Tom Carroll and calmly passed the ball into the corner of the goal before Nordfeldt had time to move.
Eriksen's second made the result safe just past the hour mark.
Tammy Abraham might have pulled a goal back with a rare Swansea opportunity, but when Vorm parried a shot from Olsson, Abraham's diving header flew straight back to the Spurs keeper.
Tottenham have reached back-to-back FA Cup semis for the first time since 1982, although they have lost all seven of those they have contested since they last won the Cup in 1991. The visitors' third goal came after a long spell of possession when Eriksen arrived on the edge of the area to thump a shot past Nordfeldt who was deceived by the bounce.