Damjanovic's winner was a further blow to the fancied Ukrainians, who had lost ground on their group rivals on Friday night after surprisingly being held to a goalless draw in Moldova.
The visitors were more aggressive from the start, threatening from a dangerous Ruslan Rotan free-kick in the first minute, then Nikola Drincic sending another set-piece into the arms of Ukraine keeper Andrei Pyatov.
Oleg Gusev had Ukraine's first chance when he shot wide in the 14th minute, and Gusev missed the target again from long range in the 36th minute.
Damjanovic forced a save out of Pyatov in the 42nd minute and three minutes later after a sustained spell of Montenegro pressure, the same player fired home from 12 yards after Stevan Jovetic's pass from the right wing.
The visitors continued to look sharp at the start of the second half, Pyatov tipping a header from Marko Basa round the post, and Stefan Savic heading just over the bar from the resulting corner.
Pyatov denied Montenegro a second goal from the penalty spot in the 77th minute after Savic was fouled by Yevgen Selin, saving Jovetic's spot-kick in superb fashion.
But the Ukrainians seldom looked like fashioning an equaliser, with Jovetic missing the target again in the dying minutes as the visitors held on comfortably for a famous win.