Cole Stockton scored the only goal of the match in the 29th minute as the Shrimps reached the third round for the first time since 2003.
Story of the match
In high winds and a downpour of rain, Buxton came out firing, Tom Elliott found himself through on goal after just four minutes but couldn’t convert.
Moments later, Jamie Ward struck an effort wide of the target at the other end, whilst Wesley McDonald had a chance, but he failed to find the target.
They were nearly in front in the 13th minute as Diego Di Girolamo hit the post with his shot from just outside of the box.
Eight minutes later, Elliott's corner appeared destined for the back of the net, but Morecambe goalkeeper Kyle Lutheran did well to turn it over the bar.
The Bucks’ goalkeeper then made a brilliant double save to deny McDonald and Stockton, before Ward fired wide of the goal just two minutes before the interval.
Substitute Ashley Chambers had a go at goal, but hit it straight at Lutheran and at the other end, McDonald had an effort off target and Stockton then fired over.
With 13 minutes left, McDonald nearly took advantage of defensive confusion before Stockton saw his shot held by ‘keeper Richardson.
In the closing stages, Stockton and Jonah Ayunga could have doubled the Shrimps’ lead, but in the end they had already done just enough to progress into the next round of the competition.
Man of the match: Cole Stockton
The forward grabbed the match-winner for his side and was a constant threat to the opposition throughout the entire match. He linked up well with McDonald to form a real presence in the final third, they will be hoping to do the same in the next round.