Newport entered the competition riding a two-game winning streak and posed an early threat with their towering 6'9" striker Kyle Hudlin.
Andy Cook's header was saved by Nick Townsend at the other end but Newport should have gone ahead after 32 minutes. Bobby Kamwa's cross picked out Aaron Wildig at the far post but the skipper scuffed the chance of a fifth goal in seven games.
Townsend saved well from Jamie Walker but had no chance with Bradford's 39th-minute goal, Cook angling a powerful header home from Jay Benn's cross.
Newport began the second half on the front foot as Hudlin rattled a 20-yarder just wide and Kieran Evans' flicked header flashed past a post.
TrendingThe equaliser came after 65 minutes with Evans picking out Hudlin to head past Sam Walker.
But Jamie Walker struck 10 minutes later, converting Bobby Pointon's cross with the help of a deflection.
And sub Pointon claimed a third after being set up by Benn.
The result lifts Bradford up to ninth in League Two, ahead of Newport on goal difference.