Calafeteanu hit three more unanswered penalties in the next 20 minutes to give his side a 12-3 lead towards the end of the first half.
However, the Falcons reduced that deficit to just two points when Adam Powell went over as the clock hit 40 minutes. Hodgson converted the effort to send the sides in 12-10 at half time.
Bucharest held firm for much of the second half and were relieved to see a Hodgson penalty hit the post with just 10 minutes left to play.
Newcastle did eventually get another chance, however, and this time they took full advantage as Tom Catterick kicked a penalty in the 76th minute to secure the narrowest of victories for the visitors.