Waverley drew first blood by sailing into the lead in the ninth minute, when new signing Jason Inglis, formerly of Selkirk and Gala Fairydean Rovers, rattled home a shot.
Just past the half-hour mark, Jordan Yardley put the home side further ahead.
A good Stow side took all of this in their stride though, and reduced the leeway before the interval, when Hagan Steele made no mistake with a penalty after Jamie Richardson had tripped an attacker in the box.
Boosted by this, Stow began to make their mark and Steele slotted home the equaliser.
From a side that could well have been dead and buried, Stow were alive and kicking and, with two minutes remaining, grabbed what was to be a winning goal.
Waverley keeper Ali Willison parried and the ball broke loose to an eager Stow man, who made no mistake on shooting home.