Elito put the visitors a goal ahead after 10 minutes when he collected a through-ball and lofted a finish beyond the reach of Ian McLoughlin.
Luke Norris then saw his effort shave the top of the crossbar as the away side looked for a second goal.
Ten minutes before the break, Newport were gifted an opportunity to level the scores when the referee pointed to the penalty when Norris handballed, but Chris Zebrowski smashed his spot kick against the bar.
The miss proved costly, as Elito doubled his tally two minutes before the break with a cool finish from close range.
Newport did get a goal back on the stroke of half time, when Robbie Willmott stroked a curling free kick beyond the reach of Chris Lewington.
The decision to award a free kick seemed to upset the visiting manager Wayne Burnett, who was sent to the stands by the referee at half time.
The second period didn't see the same quality as the first half, as both teams struggled to create many chances for their forward players.
However, Newport substitute Adam Chapman engineered a goalscoring opportunity with 10 minutes left following a neat one-two, but he sent his shot well wide of the post when well placed.
The home side's chances of securing a point were hampered with four minutes remaining when David Pipe was shown a red card, and the visitors held on to secure all three points.