Spurs will now face Bayern Munich or Fenerbahce in Wednesday’s final of the pre-season tournament.
Real put in a laboured performance in a sedate match, with £150m signing Eden Hazard at fault as they gifted the ball to Kane to slip past Keylor Navas.
Spurs, bright in the first half, were much changed for the second period and comfortably held off uninspiring Real.
Real will play again on Wednesday, meeting the loser of Tuesday’s second match in the Allianz Arena between Bayern and Fenerbahce.
Hazard peripheral and at fault for goal
There were reports he arrived at the Real camp “7kg overweight” and while the Belgium international could not be accused of lacking commitment in his 80 minutes against Spurs, he was a peripheral figure.
Playing in a second-striker role, as opposed to the wider position he used to adopt for Chelsea, he found space hard to come by and lacked sharpness in his distribution.
His most notable misplaced pass led to the Spurs goal, forcing Marcelo to stretch for the ball and inadvertently play in Kane to score past Navas.
Who will replace Trippier at right-back?
Serge Aurier is sidelined with a hand injury, collected on Africa Cup of Nations duty with Ivory Coast, leaving academy product Kyle Walker-Peters and Argentina international Juan Foyth as the remaining candidates for the right side of defence.
They played a half each in Germany, with Foyth looking comfortable in the first 45 minutes, particularly defensively as he neutralised the dynamism of wing-back Marcelo, never looking like he would be beaten one-on-one by the Brazilian.
Walker-Peters, on as one of nine half-time substitutes for Spurs, would be expected to offer more in the attacking third than Foyth, but was rarely given the opportunity to roam forward with Real on the front foot for much of the second half.
Foyth, conventionally a centre-half, would look to be in pole position to take the right-back berth in the longer term, having played there for Spurs at times last season and also for Argentina at this summer’s Copa America, but is suspended for the first two games of the season.