Kane was the star man at Wembley, scoring twice after Son Heung-min's early strike had been cancelled out by a curling long-distance effort from Andriy Yarmolenko.
His double secured Spurs a 3-1 victory and three crucial points in Group H, where Dortmund are likely to be their chief rivals for qualification to the knockout stages, in between Real Madrid and APOEL.
Mauricio Pochettino's men had it all to prove, not only at Wembley, where they had won just two of their 12 previous fixtures, but at Europe's top table, following their disappointing campaign last year.
Kane, who now has four goals in as many games in the Champions League, claims their opening win will make other teams sit up and take notice.
"Everyone knows Dortmund are one of the best teams in Europe," Kane said.
"When they see 3-1 - and it probably could have been four or five, we had a couple more chances - definitely it's a statement, and that's what we wanted to do.
"We're in this competition to try and get very far in it. It's a tough group but a night like this will only help us."