The visitors squandered a golden opportunity to take the lead midway through the first half when Gabriel Torje broke through the home defence and teed up Keseru, but the forward's shot was well over the crossbar.
The hosts, trying to reach a major tournament for the time since the 1986 World Cup in Mexico, began the second period as brightly as they finished the first, with Stuart Dallas inches away from netting his second international goal with a fizzing shot from the edge of the box.
However, despite the home side's industry and growing confidence, they nearly fell behind after an error from Gareth McAuley, whose misplaced pass allowed Torje to sprint at goal before his shot took a deflection off Evans and dribbled narrowly past McGovern's post.
Midway through the second half, Chris Brunt was unable to take advantage of a free kick in a dangerous area as he smashed his effort well over Tatarusanu's crossbar.
Late on, Kyle Lafferty almost snatched a dramatic victory with an instinctive shot, which the Romanian keeper saved comfortably.
Moments later, Oliver Norwood's free kick forced Tatarusanu into another save as Northern Ireland appeared to be finishing the stronger.
However, Michael O'Neill's team ended with only a share of the spoils, which leaves them two points clear of third-placed Hungary, with only the top two teams guaranteed a place at next year's finals in France.