Spurs, with striker Harry Kane fit again after a hamstring injury, showed intensity and quality to make their mark on the Champions League after going out at the group stage last season.
Dele Alli gave Spurs the lead after 27 minutes when he turned in Kieran Trippier's cross from close range, then doubled the advantage early in the second half with a shot that took a deflection off Real Madrid captain Sergio Ramos.
Christian Eriksen provided a cool finish from Kane's pass for the third and even though Cristiano Ronaldo pulled one back for Real with 10 minutes left, it was too little, too late, as Spurs made a powerful statement about their European aspirations.
The only blip for Spurs was the injury to defender Toby Alderweireld who came off with a hamstring injury in the 24th minute. The Belgian said he would be "out for a few weeks".
They lead Group H with 10 points, three more than second-place Real while both Borussia Dortmund and Apoel Nicosia have two points after they drew 1-1.
Tottenham's success also came on a excellent night for English clubs in Champions League action with Manchester City winning 4-2 at Napoli and Liverpool recording a 3-0 victory over Maribor.