Reus has been in stunning form since returning from injury last month but his efforts were not enough to move Dortmund back into second place.
Schalke's earlier victory over Hertha Berlin sees it jump ahead of its fierce rival, although both are well behind runaway leader Bayern Munich.
It could have been a lot worse for Dortmund after Jean-Kevin Augustin broke the deadlock before the half hour mark, the striker latching onto Naby Keita's throughball.
Dortmund, which had seen two goals ruled out for offside, restored parity before the break thanks to Reus, the forward on hand to beat Peter Gulacsi with a calm finish.
Finding a winner proved impossible for either side, though, as they were forced to settle for a share of the spoils, a result that sees Dortmund extend its unbeaten run under Stoger to 10 league matches.
Both goalkeepers were in action early on, Roman Burki denying Timo Werner before Gulacsi was equal to Michy Batshuayi's venomous effort at the other end.
Keita's return from a hamstring injury provided a timely boost for Leipzig, though the Liverpool-bound midfielder could have cost his side dear in the 16th minute when his loose pass allowed Reus to counter.
Batshuayi thought he had given Dortmund the lead from the resulting attack, only for the linesman's flag to cut short the Belgian's celebrations, Andre Schurrle having strayed offside in the build-up.
But Keita was quick to atone for his error.
Having dispossessed Schurrle with a fine challenge, the 23-year-old brilliantly picked out Augustin, who kept his composure to curl a finish into the bottom-right corner.
Leipzig's lead would not last long, however, Reus getting the better of the hosts' high-line to latch onto Mahmoud Dahoud's throughballl and slot home.
Keita's indifferent return to action continued after the interval, as his sluggish pass forced Gulacsi to rush out and prevent Batshuayi getting through on goal.
Marcel Sabitzer was fortunate not to concede a free-kick on the cusp of Leipzig's area just prior to the hour, while Emil Forsberg also escaped a booking for a lunge on Lukasz Piszczek.
Dahoud should have had another assist when he prodded a cross into Batshuayi, but the Chelsea-loanee somehow failed to apply the finish from three yards out.
Bruma's introduction injected pace into Leipzig's play, though, as the hosts managed to gain a foothold as the half wore on.
The winger could have snatched it late on when he created space in Dortmund's area, only to loft his cross-cum-shot just wide as Dortmund held firm to claim a point that keeps it in the top three.