Kane is spending a second spell on the sidelines with an ankle ligament injury, which forced Mauricio Pochettino to start Heung-min Son up front against the Saints.
But Spurs coped well without their talisman and raced into a two-goal lead before the break thanks to Eriksen's long-range effort and Dele Alli's confidently dispatched penalty.
The visitors rallied in the second-half and set up a nervous finale following James Ward-Prowse's close-range finish.
However, the north Londoners dug in to see out the game - with substitute Vincent Janssen as their central striker - and cut Chelsea's lead at the top of the table back to 10 points.
Eriksen was delighted to help Tottenham notch up a 10th consecutive top-fight victory and says it was crucial other players stepped up to the plate with Kane unable to feature.
"Everyone felt responsibility without Harry [Kane]. We had 11 players and we were good enough to substitute," he told Sky Sports.
"It was important someone else scored and we did. We had exactly the same system but with [Heung-min Son] Son up front instead of Harry.
"He is a different player but I think we did really well in the first half," the Dane insisted.