The 16-year-old overcame Zak Barton of England 3-1 after recovering from losing the first frame on the pink by securing the second frame on the green.
A tight third frame was to prove the turning point in the match when Leslie potted a good blue and pink to win take it. Leslie then went on to secure the match with an excellent 64 clearance, which is the highest break of the tournament so far.
He is due to play Benjamin McCabe of Sweden tomorrow in his second group qualifying match.
Leslie's 15-year-old compatriot, Scott Donaldson from Perth, is also competing in the event and had a comfortable 3-0 victory over Christopher Paplow of Malta.