While Pochettino will be relieved to get the win his Everton counterpart Ronald Koeman -- who gave Wayne Rooney a start despite his displeasure at the former England star's arrest for drink driving -- may begin to feel the pressure with their second successive defeat.
Kane struck on 28 minutes in bizarre fashion after Kieran Trippier's mis-hit left-wing corner had reached him at a wide angle on the right.
The England forward, who since he became a Spurs regular has failed to score in August, spied Jordan Pickford off his line and what appeared to be an attempted cross drifted over the goalkeeper into the top corner.
It was Kane’s 100th goal for the club and continued his remarkable pattern of barren Augusts being followed by almost relentless productivity.
Eriksen scored an equally scrappy effort after 41 minutes that put the visitors firmly in control after Cuco Martina failed badly to clear Dele Alli’s far-post cross.
Full-back Ben Davies saw his initial shot saved by Pickford but Eriksen responded first to pounce on the rebound and convert into the gaping net.
Koeman made two half-time substitutions but any hope of a comeback was removed after just 47 seconds when Kane scored his second.
There was again a distinct lack of quality defending as Davies was allowed the space to curl in a cross which Kane met with an emphatic side-footed finish from no more than six yards.
Strangely, the game had started brightly for Everton and Rooney.
After six minutes, he linked excellently with full-back Martina who ran onto Rooney’s return pass and his driven, near-post cross was turned just wide by Spanish striker Sandro.
Everton striker Wayne Rooney looks on during the English Premier League football match between Everton and Tottenham Hotspur at Goodison Park in Liverpool, England on September 9, 2017
But as the half wore on, Tottenham began to carve out the better chances with Alli setting up a shot for Eriksen, who drove it just wide, and Moussa Sissoko, starting just his fourth game of the calendar year for Spurs, having an effort blocked.
Matters got worse for Everton with the second goal and, before the interval, Alli almost connected with a Kane cross from six yards and the latter shot narrowly wide from the edge of the area as the home crowd responded by booing off their team.
Their mood turned ever more ugly with Spurs making such a devastating start to the second half and it took impressive reflex blocks from Pickford to keep out Eriksen and Kane in the minutes that followed, as Tottenham threatened to run riot.
Sissoko, also, should have marked a rare start with a goal but mistimed his header from yet another unopposed Trippier cross.