The competition's last remaining Highland League side held their League One opponents for over an hour and had been the better side in the opening period.
William West's early effort was saved by Chris Smith while Scott Barbour fired wide.
But it was the favourites who took the lead in the 63rd minute when McNeil curled a shot into the bottom corner beyond Paul Leask's dive.
As the visitors pushed forward for an equaliser, McNeil took advantage of the gaps in defence to add a second from close range with two minutes left.
And Errol Douglas gave the scoreline a flattering look in the last minute, receiving Bryan Hodge's pass before slotting past Leask.
- Sports & Recreation
- Scott Barbour