Harriers finished above County in the Conference last season but it was the Welsh side who earned promotion to the Football League through the play-offs.
However, it was Kidderminster's day in the cup as Gash and Gittings had the home side 4-0 up before Robbie Willmott struck twice in four minutes for the visitors late on.
Steve Burr's boys took the lead after 19 minutes when Gash guided the ball into the corner of the net with the outside of his boot from 15 yards.
Gittings struck his first just before the half-hour mark when he fired in after Lenny Pidgeley had saved Joe Lolley's effort.
It was 3-0 on the stroke of half-time when a classic counter-attack was calmly finished off by Gittings as he ran through one-on-one with Pidgeley, and Gash grabbed his second of the game with a low free-kick after 63 minutes.
Willmott scored twice with the clock ticking down when he reacted quickest inside the box to pull one back in the 79th minute before rising highest at the back post to direct a header into the far corner of the net after 83 minutes - but it was too little, too late for Newport.