The Shakers claimed just their second clean sheet of the season, but they also drew a blank at the other end, Tom Soares being the biggest culprit.
The point edged Bury up to 18th place in the table while Newport are ninth after losing once in their last eight league games.
Bury handed a first start to Marco Navas, brother of Manchester City winger Jesus, and he sliced an early shot wide after cutting in from the right.
Anton Forrester's cross then found Navas unmarked, but the Spanish winger headed wide when he could have brought it down.
Newport threatened with Danny Crow hooking the ball wide from an Adam Chapman corner, while Lee Minshull's header was cleared from under the Bury crossbar.
Forrester headed over shortly after the break and then Chapman shot too close to Bury keeper Brian Jensen.
Soares then scuffed a shot straight at Newport keeper Lenny Pidgeley before turning Navas' low cross just wide and volleying over from Shaun Beeley's centre.
Newport went closest to a late winner, with Minshull's header tipped over by Jensen before Chapman fired wide.