The France defender's goal made up for an earlier slip that allowed Pierre-Emerick Aubameyang to cancel out Cristiano Ronaldo's 17th minute opener towards the end of the first half.
Dortmund and Real each have four points with the Germans top on goal difference, while Sporting Lisbon are third on three points after beating visitors Legia Warsaw 2-0.
"My players played very well and I think we deserved a little more but this is not a bad result," Real coach Zinedine Zidane told a news conference after a third successive draw following two in la Liga.
"We suffered a little because Dortmund pressed us a lot, although we knew we were coming to a difficult ground against a difficult opponent.
"The annoying thing is we have now drawn three games but we're going to turn this around. We're not going to stop."
ELECTRIC START
The hosts made an electric start at the Westfalenstadion after Ronaldo tested goalkeeper Roman Buerki with an early free-kick as Gonzalo Castro and Aubameyang each shot straight at Real stopper Keylor Navas and Ousmane Dembele fired over.
Just as a Dortmund opener looked inevitable, Real struck from a perfectly executed counter-attack involving Karim Benzema and Gareth Bale which Ronaldo finished off by arrowing the ball into the far corner.
Thomas Tuchel's side stuck with their game plan of pressing Real high and hitting them with lightning speed, although their equaliser came from a free-kick.
Navas, making his first appearance of the season for Real, weakly punched Raphael Guerreiro's shot at team mate Varane and the ball trickled towards the goalline where it was helped across by Aubameyang.
The home side kept up the tempo after the break, although Real also grew in confidence.
Bale and Benzema wasted chances before Varane put Zidane's side ahead again, tapping home after Benzema had crashed a shot against the bar.
Dortmund again raised their game, though, and while Navas turned away two shots from lively substitute Emre Mor, he was powerless to keep out Schuerrle's rocket.
(Reporting by Richard Martin; editing by Ken Ferris and Ian Chadband)
