After a bright start from the home team, Stockton scored just before the half-hour mark to steer the Shrimps through a tricky tie played in driving wind and rain.
Buxton, who began the game 77 places below Morecambe in the pyramid, started the game on the front foot as they pressed and harried to forge early chances.
Striker Diego De Girolamo went close to giving the home side the lead in only the 10th minute.
He let fly with an effort from the edge of the area that beat the recalled Kyle Letheren but hit the left-hand post.
Former Northern Ireland international Jamie Ward looked bright and fired wide on two occasions before forcing Letheren into a near post save.
After a slow start, the Shrimps began to find their way into the game and took the lead with their first real opening on 28 minutes.
Buxton failed to clear their lines effectively and Stockton took full advantage to get beyond his marker and drill a low shot past Theo Richardson for his 16th goal of the campaign.
Richardson then denied the Shrimps a second 10 minutes later when he produced a superb save to deny Alfie McCalmont from close range.
The second half saw both sides initially struggle to create any clear chances.
Morecambe were the first to threaten when Wes McDonald drilled a low effort wide on 61 minutes, and from there, the chances came at regular intervals for the Shrimps.
Stockton twice fired over from clear openings and Jonah Ayunga failed to find the target from close range when he looked certain to score.
When Stockton did hit the target, he was denied by a superb save from Robinson low to his right.
With just one goal in the game, Buxton looked to push forward but were unable to find their way through a resolute Morecambe defence as Stephen Robinson’s side sealed victory with a professional performance.
Buxton: T Richardson, Curley, Fox, Middleton, Hurst, Meikle, Ward, Dawson (Tear, 81), De Girolamo (Chambers, 47), Elliott, Clarke. Subs not used: Milnes, Hinds, H Richardson, Dillingham, Catt.
Morecambe: Letheren, McLaughlin, O’Connor, Delaney, Leigh, McCalmont (Jones 67), Diagouraga, McLoughlin, McDonald, Stockton, Gnahoua (Ayunga 67). Subs not used: Andrésson, Obika, Phillips, Cooney, Gibson, Duffus, Wootton.
Referee: Lee Swabey.
Thanks for reading. If you value what we do and are able to support us, a digital subscription is just £1 per month for the first two months. Try us today by clicking here