Kane struck in the first minute, and Christian Eriksen scored a 71st-minute winner as Mauricio Pochettino's men moved within two points of fourth-placed Southampton.
It marked the fourth consecutive fixture in all competitions Kane has scored for Spurs, as his purple patch continued.
The England Under-21 international said UEFA Champions League qualification was still the aim of the White Hart Lane side.
"We just want to push on and build on the momentum we have gained this Christmas, keep on getting wins and see where it takes us," Kane said.
"Top four is wide open at the minute. There are a lot of teams in the hunt so we just need to keep performing and keep focused, get some wins and see where it puts us in the table.
"We've definitely been playing better in recent weeks. In the last couple of months we've gotten used to the way the manager wants us to play and I think it's coming together."
Pochettino said Kane and Eriksen - four goals in his past seven league outings - were hungry to score goals.
"I am happy for them and I am happy for the team. It was a very good result," he said.
"The reason they both keep scoring is they always try to get into scoring positions and they have the desire to score.
"It is not one of those things - they train hard for this.
"They both keep wanting to score.
"Christian has a free role. I think he is a player you need to give confidence."
Tottenham have won four straight matches, all competitions, but host Manchester United and Chelsea on Sunday and New Year's Day respectively.
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