The first concrete chance of the game fell to the hosts just after the quarter-hour mark when Constantin Budescu sent a low drive at the bottom left corner, but Orestis Karnezis pulled off a smart diving save.
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Neither side was able to break the deadlock by the half-time mark, but Gabriel Torje came close two minutes after the restart when his first-time volley flew narrowly wide of the left post.
Alexandru Maxim's free kick just before the hour was met by the head of Dragos Grigore, the ball soaring harmlessly above the Greek bar, while Torje forced another save out of Karnezis 74 minutes in after darting into the box.
Despite Romania's overall dominance, the visitors were able to survive until the final whistle in a result that, alongside Northern Ireland's 1-1 draw with Hungary, keeps things as they were in Group F.