Leverkusen would break through moments later in the eighth minute, though, thanks to Lars Bender's close-range header.
The visitors were level on the half-hour mark when Reus crossed for Oliver Kirch to finish.
They were back in front for just a few minutes, though, as Reus tucked home a penalty after Roberto Hilbert handled the ball in the area.
There was less goalmouth action after the break, with Leverkusen coming closest to scoring a winner through Son Heung-Min and Brandt.
A point sees Leverkusen edge one clear of Wolfsburg in the race for the top four, while Dortmund are now seven points clear in second.