David Pipe went close for bottom club County in a scoreless first half, but after the break eighth-placed Barnet took control.
Goalkeeper Joe Day was County's hero, saving brilliantly to foil Simeon Akinola and making an outstanding double stop to deny Mauro Vilhete.
John Akinde fired wide in the dying minutes for The Bees.
Newport County manager Graham Westley told BBC Radio Wales: "Before you become a winning side you've got to be an unbeaten side, a side that's difficult to play against and proud of its clean sheets.
"And I think in the past couple of games you've seen a side that's worked very hard at its defensive aspects.
"When you go away from home you're going to be under the cosh to an extent, but I thought it was a very determined and a very dogged performance."
Match ends, Barnet 0, Newport County 0.
Second Half ends, Barnet 0, Newport County 0.
Attempt missed. Mauro Vilhete (Barnet) right footed shot from outside the box is high and wide to the right.
Attempt missed. Josh Sheehan (Newport County) right footed shot from the right side of the box is close, but misses to the right.
Foul by Mauro Vilhete (Barnet).
Josh Sheehan (Newport County) wins a free kick in the attacking half.
Alex Samuel (Newport County) wins a free kick on the left wing.
Foul by Ricardo Santos (Barnet).
Attempt blocked. Elliot Johnson (Barnet) left footed shot from the left side of the box is blocked.
Attempt missed. John Akinde (Barnet) right footed shot from the left side of the box is close, but misses to the left.
Delay over. They are ready to continue.
Substitution, Newport County. Mickey Demetriou replaces David Pipe because of an injury.
Delay in match David Pipe (Newport County) because of an injury.
Michael Nelson (Barnet) wins a free kick on the left wing.
Foul by Aaron Williams (Newport County).
Attempt missed. Nana Kyei (Barnet) right footed shot from outside the box is high and wide to the right from a direct free kick.
Curtis Weston (Barnet) wins a free kick in the defensive half.
Foul by Alex Samuel (Newport County).
Attempt blocked. Jack Taylor (Barnet) right footed shot from the centre of the box is blocked.
Corner,  Barnet. Conceded by Sid Nelson.
Corner,  Barnet. Conceded by Joe Day.
Attempt saved. Nana Kyei (Barnet) right footed shot from outside the box is saved in the bottom right corner.
Attempt saved. Josh Sheehan (Newport County) right footed shot from more than 35 yards is saved in the top centre of the goal.
Substitution, Barnet. Luke Coulson replaces Harry Taylor.
Foul by Simeon Akinola (Barnet).
Alex Samuel (Newport County) wins a free kick on the left wing.
Attempt blocked. Simeon Akinola (Barnet) right footed shot from outside the box is blocked.
Mark O'Brien (Newport County) is shown the yellow card for a bad foul.
John Akinde (Barnet) wins a free kick in the defensive half.
Foul by Mark O'Brien (Newport County).
Foul by Elliot Johnson (Barnet).
Alex Samuel (Newport County) wins a free kick on the left wing.
Attempt missed. Mauro Vilhete (Barnet) left footed shot from the right side of the box is close, but misses to the right.
Foul by Elliot Johnson (Barnet).
Alex Samuel (Newport County) wins a free kick on the left wing.
Attempt missed. Mauro Vilhete (Barnet) right footed shot from outside the box is close, but misses to the left.
Corner,  Barnet. Conceded by Joe Day.
Attempt saved. Mauro Vilhete (Barnet) header from very close range is saved in the centre of the goal.
Simeon Akinola (Barnet) wins a free kick in the defensive half.
Foul by Tom Owen-Evans (Newport County).
Newport County survived a late Barnet onslaught to claim a second successive draw in League Two.