The Lilywhites had chances to win the game, Sean Maguire hitting the bar and Emil Riis ballooning the ball over the bar from close range.
Frankie McAvoy now takes his side to face Sheffield United at Bramall Lane on Tuesday night.
Here are the PNE player ratings from the clash in the West Country, with Daniel Johnson the pick of the bunch for the visitors.
1. Daniel Iversen 8
Made three saves as Bristol City tested him with shots from outside the box. Also did well to smother at Chris Martin’s feet.
2. Jordan Storey 7
Steady game in the back three bar an early misplaced header, dealt with Martin when the experienced striker tried targeting him.
3. Patrick Bauer 7
Little got past the German defender with him handling the threat of Weimann well. Used his experience well.
4. Andrew Hughes 7
Another tidy game on the left side of the back three. Had a late header saved at the other end of the pitch.