The stalemate keeps Leverkusen second in the group, still a point away from leaders Manchester United who were held to a goalless draw at Real Sociedad.
Leverkusen had goalkeeper Bernd Leno to thank for securing a valuable away point, making a string of saves.
The Ukrainian side dominated in front of their own fans at the Donbass Arena, chalking up 17 opportunities, including nine shots on goal, only for Leno to keep them at bay.
Donetsk, who remain two points behind Leverkusen, may live to regret their failure to beat Leno, who was in inspired form as he produced a string of fine saves.
Leno set the tone for his performance, and the match, with two stops in the opening 12 minutes to deny Facundo Ferreyra, with the second an acrobatic effort.
Just after the half-hour mark Leno was beaten for what proved to be the only time in the game, only for a deep cross from Darijo Srna to fortuitously hit a post.
Leno then followed up with saves from Brazilian midfielder Bernard and Yaroslav Rakitskiy, the latter an instinctive stop to a close-range volley.
The Germany Under-21 international then started the second half as he finished the first, pulling out another top-drawer save to keep out a long-range free-kick from Srna in the 47th minute.
Although Leverkusen went on to enjoy a stronger second half, they were again indebted to Leno in the dying stages in thwarting Luiz Adriano with a header.