Forward Mikhail Gordeichuk gave the home side a second-minute lead before Admir Mehmedi replied midway through the second half with a superb finish.
The outcome left Leverkusen in third on five points from as many games, a point ahead of BATE.
Both Leverkusen and BATE will be eliminated before the final round of games if AS Roma beat holders Barcelona at the Nou Camp later on Tuesday.
The action-packed draw in Belarus also sent Barcelona, who top the section with 10 points from four games ahead of Roma with five from four, through to the last 16.
Leverkusen dominated throughout but poor finishing let them down after BATE scored with their first move, capitalising on a calamitous error by Leverkusen goalkeeper Bernd Leno.
Gordeichuk took the ball in his stride on the left flank and cut inside two defenders before unleashing a low shot which trickled into the net through Leno's arms and legs.
Bayer pressed on relentlessly for the remainder of the first half but failed to threaten until the 45th minute, when their top scorer Javier Hernandez missed a gilt-edged chance.
Hernandez was denied by a reflex save from goalkeeper Sergei Chernik, who kept out the Mexico striker's close-range shot with his feet after good work by Mehmedi.
The visitors piled on the pressure after the break as Hernandez fired inches wide from long range and Karim Bellarabi had a shot blocked after a dazzling solo run.
Leverkusen finally drew level when livewire Mehmedi side-footed home a delightful first-time shot that went in off the far post after a good pass by playmaker Hakan Calhanoglu.
The German side laid siege to the hosts in the closing stages but could not find a way through a well-organised BATE defence.
(Writing by Zoran Milosavljevic in Belgrade; editing by Toby Davis)
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