The Spanish top seed dropped the first set before recovering to level and then led by a break in the decider, but world number 25 Lehecka won the final four games for a 6-3, 3-6, 6-4 victory.
Alcaraz entered the clash on a seven-match winning streak following his triumph in Rotterdam earlier this month but ran into trouble against a confident Lehecka, who won his second career title in Brisbane to start the year.
"I feel great," said Lehecka. "To win a match like that against a player like this is a super big achievement. I believed in myself and knew I had the level to produce this kind of tennis.
"I think the most important part was I believed in my game and did not back down."
Lehecka is through to the last four in Doha for the second time.
The Czech will play eighth seed Jack Draper for a place in the final after the Brit downed Italy's Matteo Berrettini in the day's last quarter-final, winning 4-6, 6-4, 6-3.
Earlier in the day, Andrey Rublev converted his eighth match point against De Minaur to take his place in the semi-finals.
The Russian, a Qatar winner in 2020, had to dig deep to down the Australian 6-1, 3-6, 7-6(8).
Fifth-seeded Rublev let slip a 5-2 lead in the final set as De Minaur scrambled back into contention to force a tiebreak.
Rublev finally got over the line, the 27-year-old saving a match point himself in a memorable quarter-final clash.