The result means Buxton extend their winning run in the league to three games - four across all competitions - while claiming their third succesive clean sheet in the league.
It also moves the Bucks up five places in the UniBond Premier Divison, t
o 12th, with 23 points.
Burscough are now without a win in their last six games, amid talk of off-field problems.
Elsewhere in the league, Boston United are the new leaders after winning 5-0 at Ossett Town, whose defence has suddenly sprung a leak having conceded seven the previous Saturday.
The Pilgrims replace Retford United, whose home form is letting them down and they have won only one in ten on their own patch after Kendal Town grabbed a point in stoppage time.
Bradford Park Avenue moved in to the play-off places after beating Frickley Athletic, who are still looking for a first win on the road.
Stocksbridge PS were held at home by Worksop Town, and Hucknall Town beat third place Ashton United.
The performance of the day in the FA Trophy was Nantwich Town's 3-0 victory away at Eastwood Town, while North Ferriby United, AFC Fylde and Mossley all earned draws against Conference opposition.
In the all-UniBond games, Guiseley and Kings Lynn pulled rank on FC Halifax Town and Salford City, but Witton Albion held Premier Division Matlock Town.
Don't forget to watch Talking Balls on Monday - our light-hearted video round-up of the weekend's High Peak football results. Click here to see last week's show.