The Denmark international, who picked up the man-of-the-match award, fired in a low drive from 25 yards before Dele Alli struck from the penalty spot as Spurs eased into a 2-0 half-time lead.
James Ward-Prowse pulled one back shortly after the restart and despite both sides creating several chances, the hosts managed to hold onto victory which takes them 10 points behind Premier League leaders Chelsea in second place.
Star striker Kane is facing a spell on the sidelines with an ankle injury, but Eriksen believes Spurs have enough quality in their ranks to cope without him.
Eriksen told Sky Sports: "Everyone felt the responsibility. The players are good enough to substitute (Harry Kane's goals) so obviously it was important for someone else to score.
"It is exactly the same, we play the same system but different players with Sonny (Son Heung-min) instead of Harry and he plays a bit more deep but he's a terrific player and he wants the ball at his feet.
"He's a different player and I thought we did really well but the second half was tough for us."