Eriksen capped a masterful midfield performance by scoring twice to take his personally tally against Swansea to an astonishing eight goals in 10 games, while Erik Lamela was also on target.
Lamela added a second in injury time at the end of the first half before Eriksen scored again just after the hour mark.
Tottenham, now, are heading the right way on a very exciting journey, but with the vultures circling around the best players, who could all earn far bigger money elsewhere, a winner's medal is the flawless reassurance for Pochettino's men that this is a road worth staying on, despite whatever attractions lay elsewhere. The race for Champions League qualification continues - they are among the four teams battling for three spots behind City - but the FA Cup has also suddenly become a priority.
Spurs fans were not happy with the inclusion of Moussa Sissoko.
Kane suffered ankle ligament damage against Bournemouth that is expected to keep him out until April, but on the evidence of Spurs' display at the Liberty Stadium on Saturday, they should be able to cope without him for the time being.
Eriksen had already began to drop into various unsafe positions and made the hosts pay when Spurs won the ball back deep inside their own half and broke forward quickly.
FA Cup quarter-final updates from the 12.15pm GMT gameBig interview: Swansea City's Sam ClucasAnd feel free to email your thoughts to Scott 5.43pm GMT This is Swansea City's first FA Cup quarter final for 54 years.
But after Eriksen and Moura combined, Lamela danced past Tom Carroll and swept home right-footed.
It was the ideal start for Pochettino's men and they continued to press forward and Moura cut inside an opponent with quarter of an hour played, but his tame shot was easily saved by Kristoffer Nordfeldt.
Eriksen's goals were terrific, but arguably the best thing he did all game was a brilliant pass over the top that Son superbly controlled and then struck into the net, only for the goal to be ruled out for offside (a VAR consultation confirmed he was a few inches beyond the last man).
Either side of Eriksen's wonderful 11th-minute opener and Lamela's terrific effort on the stroke of half-time, Spurs created chance after chance.
Spurs' Danish playmaker was pulling all of the strings against Swansea and almost added to his tally eight minutes before the break.
Eriksen was afforded time and space on the edge of the Swansea box and the 26-year-old let fly with his left foot. Had one shot on goal that went well wide.