The 23-year-old put the visitors ahead in the 10th minute at Rodney Parade, finding the space to lob home goalkeeper Ian McLoughlin.
Chris Zebroski could only hit the crossbar from the penalty spot as the hosts sought a response after Luke Norris' handball, before Elito made him pay a heavy price by coolly slotting home Dagenham's second.
Robbie Willmott found the back of the net with a free-kick to give Newport hope just before the break, and Dagenham suffered a further setback when boss Wayne Burnett was sent to the stands at half-time after a run-in with referee Charles Breakspear.
The second half looked set to pass without incident following a frantic opening period, but David Pipe was sent off in the closing stages after being shown a second yellow card - although Dagenham could not make their man advantage count by adding to the scoreline.
The win lifts Dagenham to 11th, while Newport now have just one victory from their last 15 fixtures.
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