Despite the nature of the win, Kane has urged the players to remain cautious and "not get ahead of ourselves" as the focus shifts to the Group G decider against Belgium next week.
When asked by BBC Sport about his thoughts on the victory in Nizhny Novgorod, the Tottenham Hotspur star replied: "Fantastic. I am so, so proud of the boys.
"We are just enjoying it - enjoying being here. We have been working on set pieces and working on the way we play and it is all coming together. Fantastic to be through and to do it this way is brilliant.
"The fans have been brilliant, both sets of fans to be fair, that is what the World Cup is about. We are enjoying it and we have to keep it going, not get ahead of ourselves."
England and Belgium, who are level on points, goals scored and goals conceded, will face off in Kaliningrad on Thursday.