Reus finished off into an empty net after Jean-Kevin Augustin had opened the scoring for the Bulls, who have just one point to their name from the last nine on offer to slip down to sixth - three points off their third-placed opponents.
Another goal was ruled out for the visitors soon after, this time after Reus put the ball into the net from an offside position, and Leipzig soon pounced as Augustin edged his side in front.
The French youngster fired home a first-time effort after Naby Keita poked the ball into his path to make it three goals in three games.
Dortmund, and Reus in particular, responded well to the setback as the forward kept his composure by rounding Gulasci and rolling the ball over the line.
A bright start to the second period almost saw last season's runners-up regain their lead, with Kevin Kampl having a couple of shots kept out.
Some nice interplay between Mario Gotze and Reus almost allowed the latter to fashion a shooting chance, though clear-cut openings proved to be harder to come by in the second 45.
The best of the remaining chances fell the way of Batshuayi, who could not convert from an unmarked position four yards out, ensuring that the Champions League chasers had to settle for a point apiece.
RB LEIPZIG (4-4-2): Gulacsi; Laimer, Orban, Upamecano, Klostermann; Sabitzer, Keita, Kampl (Ilsanker 66'), Forsberg (Bruma 61'); Augustin (Poulsen 75'), Werner
BORUSSIA DORTMUND (4-2-3-1): Burki; Piszczek, Akanji, Toprak, Schmelzer; Weigl, Dahoud; Schurrle, Gotze (Philipp 78'), Reus (Pulisic 88'), Batshuayi