Both teams passed up good chances to score but it was the visitors who created the most opportunities and they will have the advantage in Wednesday's return leg in Donetsk.
Theofanis Gekas passed up Greece's best chance with a first-half header while Ruslan Rotan was guilty of missing Ukraine's clearest opening when he dragged a shot wide.
Liverpool defender Sotiris Kyrgiakos could have won the match for former European champions Greece in the dying seconds but he flashed a header wide from a cross by Konstantinos Katsouranis.
"We now have a slight advantage because we are playing at home," Ukraine coach Olexiy Mikhailichenko told reporters.
"We tried to play on the counter attack and our game plan was effective. We now have to sit down and analyse the game to see how we will approach the return."
Both sides looked lively in the opening exchanges and it was the visitors who registered the first shot on target when Greek goalkeeper Alexandros Tzorvas stopped Andriy Shevchenko's low free kick from 25 metres.
Giorgos Karagounis replied with a strike from distance which was comfortably held by Andriy Pyatov in the Ukraine goal.
Shevchenko tried his luck again from another free kick in the 13th minute but his tame strike drifted wide.
FLAGGED OFFSIDE
Gekas, who scored 10 goals in qualifying, had the ball in the net when he converted Dimitris Salpigidis's cross only to be flagged offside.
Rotan then wasted an excellent chance for Ukraine when he dragged his shot wide from the edge of the box following a defensive mix-up.
Ukraine continued to look more dangerous, particularly on the break, and Tzorvas made a smart block from Oleh Gusev.
Greece threatened twice before halftime but Giorgos Samaras's control let him down from a Salpigidis centre and Gekas headed straight at Pyatov from a free kick.
Samaras, who drifted in and out of the game in the first half, drew another save from Pyatov as Greece started strongly after the break.
Karagounis then had to come to Greece's rescue with a saving tackle to deny Artem Milevskiy.
Tzorvas pushed away a fierce Taras Mykhalyk drive in the 68th minute before coach Otto Rehhagel gave young striker Kostas Mitroglou his debut as a replacement for Salpigidis.
The 21-year-old Olympiakos Piraeus player showed some good touches but it was Samaras who went close with 10 minutes left, firing just over the bar before Kyrgiakos's late miss.
"Ukraine were very effective with their passing combinations and their attacking players were very good," said German Rehhagel.
"If we had conceded a goal it would have made things extremely difficult for us. I think we scored a perfectly valid goal but the referee disallowed it and we are disappointed."
