Reigning men's world champions Italy had been seeking to lead Group A after a winning start against Australia, but the draw pushed them into second place behind Croatia.
Greece got their first point on the board, but lie second from bottom in fifth place.
The Greeks made the first move in the opening quarter and went on to edge the period 2-1.
Italy took the next quarter by the same scoreline and the teams went into the half-time break with three goals each.
The Italians, wearing blue caps, carved out a one-goal lead in the third quarter (6-5) to set up a crucial fourth period.
Greece, in white, clawed a goal back, before Maurizio Felugo put Italy in the lead and Christos Afroudakis levelled the score with three minutes to play.