Two Harry Anderson goals in the second half ensured Portadown progressed to the next round of the BetMcLean Cup at Newington's expense.
Although scoreless at the break both sides had their chances during the first half.
The Ports got off to a bright start with Stephen Teggart and Lee Bonis both having chances inside the opening five minutes.
As the game progressed Newington began to get a foothold on proceedings, Daire Rooney sent a free header the wrong side of an upright following quickly by a Gary Warwick shot over the bar.
Portadown had a chance right at the end of the first half but Swans keeper Marc Maybin did well to tip away George Tipton's low effort from the edge of the box.
Newington threatened again early in the second half when Rooney's close range effort was stopped by the feet of keeper Jethren Barr. They also had the ball in the net moments later but this was ruled out offside.
The deadlock was broken just before the hour. Tipton thumped an effort from outside the area off the Newington crossbar but Anderson, just moments after taking to the pitch, was on hand to pounce on the rebound and head the opener past Maybin.
The visitors were reduced to nine men on 65 minutes when, following a scuffle, both Rooney and Richard Gowdy were dismissed by referee Mark Dillon.
Portadown looked to kill the game off with Bonis heading against the bar soon after before Oisin Conaty was denied at close range by Maybin and Anderson saw an effort turned away for a corner.
But he did double Portadown's lead with nine minutes remaining when Paul Finnegan hooked the ball back across the face of goal for the 18-year-old to head home at the left post.