Willmott lit up a dour first-round replay contest at Rodney Parade with the only goal of the game on the stroke of half-time.
Braintree full-back Ryan Peters fouled Conor Washington and Willmott fired past a stranded Nik Hamman from the left edge of the penalty box for his second goal of the season.
The win for Justin Edinburgh's side sets up an away tie at Kidderminster in the second round.
Braintree held their League Two opponents to a 1-1 draw in the original tie and started brightly in the return clash when striker Dan Holman stung the palms of Newport goalkeeper Lenny Pidgeley on a chilly evening in south Wales.
The visitors continued to frustrate the hosts and should have taken the lead themselves through Matt Paine's header, but the centre-back could only direct his effort straight at Pidgeley, who hours earlier attended the birth of his second child.
Yet out of nowhere, Newport showed their quality when Willmott's free-kick deceived keeper Hamman to fly into the far corner.