Lehecka recovered from a break down in the opening set with two breaks of his own and after dropping the second set, the world number 25 hit back from 4-2 down in the decider.
Felix Auger-Aliassime clinched his place in the last four after world number six Daniil Medvedev retired due to food poisoning after losing the first set 6-3.
Draper let slip his third service game on his way to losing the first set before breaking Berrettini for the first time at 5-4 up in the second to level it up.
The British number one, sidelined for a month due to a hip injury before his return to action this week, overcame the 28-year-old Italian 4-6 6-4 6-3 and faces Jiri Lehecka in the last four.
The eighth seed broke Berrettini’s serve for the second time to move 3-1 up in the decider and that proved decisive as he then fended off break point against him to lead 5-2 before comfortably serving out for the match.
