World number 91 Mann has only previously featured once in the quarter-finals of a ranking tournament – reaching that stage of the 2021 Northern Ireland Open.
Four-time world champion John Higgins was made to sweat by world number 88 Louis Heathcote after the Englishman took the first two frames with ease.
But Higgins turned on the style thereafter, winning five straight frames – including 135 break – to win 5-2 as Heathcote scored just 39 more points.
Higgins will meet fellow Scotsman Anthony McGill in the semi-finals after he fought back from 3-1 down to win 5-3 against 19-year-old Stan Moody, who eliminated Ali Carter in the previous round.
McGill has not featured in the last four of a ranking event since the 2022 Northern Ireland Open.
Saturday’s semi-finals will be the best of 11 frames (first to six), while Sunday’s final will be the best of 19 (first to 10).
Each player has won £20,000 for reaching the semi-finals, while the runner-up will receive a further £45,000 and the winner £100,000.