The home side were quickest out of the starting blocks, charging forward as early as the second minute, but Kirill Kovalchuk's effort flew just wide of the Slovakian post.
The PAOK man was able time to steady himself, take a touch, get his head up and pass the ball into the near post to give the visitors the lead.
The start of the second half saw more of the same from the home side, as Edmar struck the post before seeing his rebound stopped immediately after.
A frantic spell also saw Viacheslav Shevchuk and Andriy Yarmolenko try their luck, but the visitors emerged unscathed.
Ukraine kept up the pressure thereafter, dominating possession and firing on goal time and time again, although Slovakia caused trouble in the 67th minute when Miroslav Stoch played a ball to Filip Kiss, whose subsequent shot flew wide.
For all their efforts, however, the home side was toothless in front of goal and, unable to breach the opposition's clean sheet, saw their qualifying campaign begin with a defeat.