Medy Elito's first-half brace lifted Dagenham two places and preserved their play-off hopes in League Two.
But manager Burnett watched the entire second half, sat alone in the away stand, following his dismissal for confronting referee Chris Breakspear.
That was despite his team leading 2-1 at the break thanks to former England Under-19 winger Elito.
His first was a cool finish over goalkeeper Ian McLoughlin from Brian Saah's ninth-minute free-kick.
Newport went close to equalising three times in two minutes but Robbie Wilmott's curling effort was tipped over by Chris Lewington.
Lee Minshull then hit the post before his header struck Luke Norris' arm but top-scorer Chris Zebroski blazed over from the penalty spot.
He was made to pay when Elito fired in his second on 43 minutes before Lewington misjudged Wilmott's free-kick in first-half injury-time.
That sparked a furious tirade from Burnett who had to be restrained three times as he confronted the officials as they left the pitch at half-time.
Newport captain David Pipe was also sent off late on for a second yellow card.