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.
Mauricio Pochettino said in his pre-match news conference that lifting the trophy would not help breed a "winning culture" at Spurs, but their challenge was boosted inside 11 minutes when the excellent Eriksen found the net in stunning fashion from just outside the area.
Although the visitors were frustrated by having a Son Heung-min goal disallowed for offside shortly before Eriksen struck the crossbar, a second goal arrived in first-half stoppage time when Erik Lamela scored in style.
Swansea responded positively at the start of the second half, but they could not find a way past their former goalkeeper Michel Vorm and Spurs took advantage just after the hour when Eriksen doubled his tally.
Any chance of a Swansea comeback was long gone as the match entered its final stages, allowing Spurs to ease to successive FA Cup semi-final appearances for the first time since 1981-82, when they won the competition.
After a tight opening 10 minutes, the indications were it was going to take a moment of individual brilliance to break the deadlock.
And that was exactly what happened, as Eriksen marched towards the penalty area and bent a left-footed effort past Kristoffer Nordfeldt from 25 yards.
The Swansea goalkeeper was beaten again in the 24th minute, but Son was adjudged to be marginally offside – a decision upheld by the video assistant referee – just before firing past Nordfeldt.
Eriksen was proving a real nuisance to Swansea and he almost doubled his tally seven minutes prior to the break, though this time his long-range, left-footed strike was tipped onto the bar by Nordfeldt.
Lamela was not to be denied on the stroke of half-time, however, as he evaded a challenge in midfield and then coolly swept home from 20 yards, finding the bottom-right corner.
Swansea started the second half in encouraging fashion and tested Vorm twice in quick succession – the Dutchman pushing away Martin Olsson's rasping drive and parrying Tammy Abraham's header on the rebound.
That was not the start of a Swansea onslaught, though, as the visitors nearly added a third when Nordfeldt just about kept Eric Dier's 30-yard drive out of the top-left corner.
But Nordfeldt was arguably at fault as Spurs went 3-0 up; he managed to get a touch to Eriksen's low shot from the edge of the area but could not keep it out.
Pochettino introduced Dele Alli and Fernando Llorente from the bench late on and the Spaniard went close to scoring towards the end during a goalmouth scramble, but his failure to do so against his former team had no impact on the outcome as Spurs march on in the FA Cup.