The third seed dominated the opening set after breaking a nervy Lehecka early on but was pushed hard after that by the Czech who has enjoyed a breakout tournament.
Lehecka, who had never won a Grand Slam match before arriving in Melbourne, used his powerful baseline game to stay with Tsitsipas in the second set but the Greek raised his game to ease through the tiebreak.
Tsitsipas was forced to save three successive break points at 3-3 in the third set as Lehecka threatened to extend the Rod Laver Arena clash.
Tsitsipas then pounced with Lehecka serving at 4-5, reaching match point with a searing backhand pass and wrapping up the victory as his opponent netted a backhand.
Tsitsipas will next face Russian Karen Khachanov (26) in the semi-final when he will again enjoy strong support from the many Greek-origin fans in Melbourne.
"I can say it was a fair dinkum type of performance," Tsitsipas said on court. "It felt different this time from the other time we played but I found a solution.
"I had to deal with groundstrokes that were coming at me heavy and deep. I put my heart out there."
Lehecka, who enjoyed impressive wins against seeded players Borna Coric, Cameron Norrie and Felix Auger-Aliassime during his run, will rue not taking any of eight break points.
His big chance came when Tsistipas double-faulted to go 0-40 down at 3-3 in the third set, but he was unable to convert any of them and after that, the writing was on the wall.
Tsitsipas loves the Melbourne fortnight and has looked in top form this year, winning four of his five matches so far in straight sets with only Jannick Sinner able to extend him in a five-set clash in the last 16.
He began aggressively against Lehecka and broke the Czech's first service game at the fourth time of asking, hoisting a lob on the stretch that proved too awkward for his opponent.
It gave him a flying start but after he took the opener there was little between the players as Lehecka, whose hard-hitting game looks to be modelled on Czech former player Tomas Berdych, pushed Tsitsipas to the limit.
He got to within two points of taking the second set but Tsitsipas was too good in the tiebreak.
Despite defeat, Lehecka's run here marks him out a player capable of flying up the rankings from his current 71.
"When I think about it, of course, I mean, if somebody has told me before the tournament that I would play quarters here, I would take it for sure," he told reporters.
"Now it's bittersweet because I can't still enjoy the success. Now it's hard for me to say how happy I am about it."
