Goals from Padraig Amond and Mickey Demetriou were enough to overcome Harry Kewell’s side.
Crawley captain Jimmy Smith pulled a goal back in the early stages of the second half but it was not enough to overcome Newport.
Amond put the Exiles into the lead when he latched on to a cross from Ben White with a header from 10 yards out.
Newport extended their lead a minute before the interval with a penalty from Demetriou after Josh Sheehan was fouled in the penalty box.
Prior to the goals the Exiles came close on two occasions with both Amond and Sheehan hitting the crossbar. Smith pulled a goal back straight after the interval when his volley found the back of the net.