Dylan Mottley-Henry scored either side of a McCourt long-range effort to secure another 3 points on the road for The Bucks.
The Turbines had the first chance of the game from a free-kick opportunity. Connor Brown took out Kaine Felix on the edge of the box, and Hugh Alban Jones hit his shot low, looking for the bottom right-hand corner with Joe Young forced to deny the forward from opening the scoring.
Buxton came straight back at the hosts, with Joe Ackroyd being involved twice in the attack. Firstly, a layoff from Tommy Elliott to him was blocked away, then a cross looking for Mottley-Henry was caught by Peter Crook in goal.
Felix was continuing to be a big threat, latching onto a loose ball in the 15th minute and forcing The Bucks goalkeeper into a diving save.
The home side was able to take the lead in the 25th minute. A quick counter-attack following a Bucks corner with Felix intercepting the ball and finding Elicha Ahui who passed the ball to Dan Jarvis. With a deflected shot, it caught Young out and gave them a 1-0 lead.
And they almost scored again, with Gash breaking through and getting a shot off, only this time it came safely into the hands of The Bucks goalkeeper.
However, it was The Bucks grabbing the next goal and an equaliser in the 31st minute. Elliott found Diego De Girolamo, and despite Crook pulling off a great save, Mottley-Henry was able to put the rebound through his legs for 1-1.
Young had to be quick off his line once again, stopping Felix from lifting his shot goalwards.
And with a wonder strike, McCourt gave the visitors the lead in the 39th minute.
McCourt, from 25 yards out, hit a rocket of a shot which cannoned off of the bar and into the net to give The Bucks the lead at the break.
Peterborough Sports started the second half the brighter of the sides, winning corners and Micheal Gash attempting to dink Young in goal.
Elliott on the left-hand side had some great footwork and was able to get the ball to Ackroyd, and his shot was stopped out for a corner. The corner was met by the head of Hunt, with a Peterborough defender clearing it off of the line.
Buxton tried a free-kick routine that almost found the back of the net after De Girolamo’s smash of a shot was tipped over by The Turbines keeper.
Peterborough Sports thought they should’ve had a penalty when their forward was taken down, but the referee waved away the shouts.
Mottley-Henry added his second in the final minute and sealed all 3 points for The Bucks in the final minute. Adam Livingstone crossed the ball into the box, and Crook was unable to get a clean punch on the ball as Mottley-Henry finished it off for 3-1.