The result left both teams, who entered the game with 100 per cent records in the pool, locked together on seven points along with Slovakia after three matches.
The pressure to avoid defeat was evident in the opening stages but the home side made an early impression with Theofanis Gekas striking the post with less than five minutes on the clock.
Bosnia-Herzegovina goalkeeper Asmir Begovic also had to be alert to save a long-range free-kick from Vassilis Torosidis in the 22nd minute as the home side continued to dominate the early stages.
Greece keeper Orestis Karnezis was not tested until the 26th minute, when he held a shot from Edin Dzeko, but the visitors almost grabbed the lead on 41 minutes when Karnezis produced a fantastic save to deny Senad Lulic.
The visitors started the second half well, with Dzeko flashing a 49th-minute shot just wide, but Begovic was forced to punch away a succession of corners as the Greeks pushed to break the deadlock.
Begovic saved with his feet to deny Dimitris Salpingidis in the 79th minute, while at the other end Karnezis caught a curling 25-yard strike from Miroslav Stevanovic.
Neither side looked particularly likely to grab a winner in the dying minutes, Zvjezdan Misimovic coming closest in the 87th minute with a dipping corner that was well cleared by Alexis Tsiolis.