An own goal from Raphael Varane had seen Tottenham take the lead in the 28th minute, but Cristiano Ronaldo netted from the penalty spot late in the first period to ensure that both teams moved onto seven points in Group H.
Gareth Bale's absence against his former club meant that Isco joined Ronaldo and Karim Benzema in the final third, while Keylor Navas was fit to return between the sticks after missing the clash with Getafe at the weekend.
As for Tottenham, summer signing Fernando Llorente started alongside Harry Kane in the final third, while Harry Winks kept his position in the team due to Dele Alli's suspension and injuries to Victor Wanyama and Mousa Dembele.
It was the visitors that had the first half-chance of the match inside three minutes, but after turning Varane inside the Real Madrid box, Llorente could not produce a strike on goal.
Real Madrid then had two excellent opportunities down the other end, but Ronaldo's header from a Hakimi cross hit the post, before Benzema fired the rebound wide of the post as Tottenham survived two dangerous moments.
The home side's next chance arrived in the 17th minute after brilliant combination play between Luka Modric and Benzema, but Ronaldo sent his low effort just wide of the far post.
Kane's first sniff of the Real Madrid goal arrived in the 19th minute when the England international met a corner from Christian Eriksen, but the striker's point-blank header was kept out by Navas.
Tottenham continued to threaten on the counter-attack as the first period progressed, with Moussa Sissoko using his pace in wide areas to trouble the home side.
Indeed, it was through that avenue that Spurs made the breakthrough in the 28th minute as Serge Aurier's brilliant cross from a wide position was flicked home by a combination of Kane and Varane, with the Real Madrid centre-back having the final touch before it landed in the back of the net.
Isco brought a decent save from Hugo Lloris just past the 30-minute mark as the home side looked to hit back, before Benzema just failed to connect on a cross from Modric 10 minutes before the break.
Real Madrid did level the scores in the 43rd minute, however, when Ronaldo netted from the penalty spot after Aurier had brought Toni Kroos to the deck with a rash challenge.
Los Blancos had a golden chance to take the lead in the 55th minute when Casemiro found the head of Benzema inside the Tottenham box, but the striker was denied by a stunning save from Lloris.
The away side's first opportunity of the second period fell to Davinson Sanchez just before the hour, but the centre-back's header from an Eriksen corner was always rising over the crossbar.
It was Real Madrid that were the dominant team, however, with Kroos just missing the post in the 60th minute, before Lloris was forced to tip a Ronaldo strike over the crossbar three minutes later.
Lloris then kept out another Ronaldo effort in the 65th minute as the pressure remained firmly on the Premier League side.
Kane had a golden chance to send Tottenham back ahead in the 71st minute after Llorente had found the England international inside the Real Madrid box, but Navas made a stunning save to keep out the striker's effort.
Navas then made an equally impressive stop to prevent Eriksen from finding the back of the net as Tottenham looked to take the lead at the Bernabeu for the second time on the night.
Neither side could find a second in the latter stages, however, with the points shared after an entertaining clash in the Spanish capital.
Next up for Real Madrid is a La Liga match against Eibar at the Bernabeu on Sunday night, while Tottenham will return to Premier League action at home to Liverpool on Sunday afternoon.