The Jets nearly found an equaliser when Andrew Hoole weaved his way past three defenders into the box, but he failed to direct his effort on target as his strike hit the side-netting.
Wellington doubled their lead just before the break when Burns broke away and was one-on-one with Birighitti. The keeper saved his first effort, but the rebound fell kindly for the striker to slot the ball home.
In the second half, the Phoenix effectively secured their victory when Burns turned provider for Michael McGlinchey, who fired the ball into the top corner, leaving Birighitti with no chance.
Newcastle nearly got a goal back when Jeronimo Neumann raided forward, but just as he was about to pull the trigger, Andrew Durante won the ball with an inch-perfect challenge.