The result keeps the Bantams in the top half, one point ahead of Newport and just three points off the automatic promotion spots in a congested League Two table.
The hosts enjoyed the majority of possession and struck the woodwork early in the second half when James Clarke’s close-range header bounced off the crossbar.
The Bantams’ best opportunity of the contest came late on but Joe Day was on hand to deny Alex Gilliead.
“We had to deal with a lot of pressure,” admitted Adams.
“In seasons gone by we probably would have lost that game but I think the mental strength was there today.
“This is a difficult venue to come to and I thought we dealt well with the whole occasion.
“I think this is the toughest game we’ve had this season,” added the former Morecambe manager.
“We haven’t come up against a team who threw everything at us like they did – they put crosses into the box and corners and free-kicks. That’s how they try to get the goal.”
A win could have thrust Bradford into sixth place but Adams felt his side lacked the key pass or shot to break the deadlock.
At the other end, Richard O’Donnell produced a fantastic first-half stop to deny Mickey Demetriou and preserve City’s clean sheet.
“I thought we started the game very well,” he added.
“We got into Newport from the start of the game and got down the sides of them, which caused them a lot of trouble.
“But, in the end, we haven’t picked the right pass or had the shot at goal to get that first goal.
“It was a bit of a topsy-turvy afternoon. I would say we had the better of the first half and Newport had the better of the second.”
Bradford City: O’Donnell; Threlkeld (Robinson 45), P O’Connor, Songo’o, Ridehalgh; Watt, Sutton, Cooke, Gilliead, Vernam (Evans 66); Cook. Unused substitutes: Eisa, Foulds, Kelleher, Staunton, Scales.
Newport County: Day; Clarke, Dolan, Demetriou; Norman, Wilmott, Upson (Azaz 45), Cooper (Fisher 84), Haynes; Telford, Baker-Richardson. Unused substitutes: Townsend, Ellison, Lewis, Cain, Abraham.
Referee: S Purkiss (London)