Following a lackadaisical start, Lee made amends when he struck form midway to snatch a come-from-behind win.
While the lone overseas cueist took time to find his rhythm, Khan grabbed the opportunity to take an early lead.
But Lee, who has a recorded break of 137, used the third frame to launch his comeback and went ahead 3-2 following a black ball finish in the fifth frame.
Khan drew level by winning the next frame to raise hopes of a tight finish in the best-of-nine frames quarter-finals.
But his challenge ended there as Lee capitalized on Khan's blemishes in the seventh and eighth frames to set up a semi-final clash with India junior No. 1 Laxman Rawat of Delhi, who beat Manan Chandra of BPCL 5-0.
Rawat fired a break of 122 for the United Phosphorous-sponsored tournament's second highest that surpassed Rafath Habib's 111.
Bombay Gymkhana tournament winner Sourav Kothari of ONGC and 2009 champion Kamal Chawla of Bhopal will contest in the other semi-final after recording identical 5-0 wins in the remaining quarter-finals.
Results: Quarter-finals: Kamal Chawla (MP) bt Sundeep Gulati (Delhi) 62(56)-24, 49-14, 75-35, 65-43, 76-0; Sourav Kothari (ONGC) bt Alok Kumar (ONGC) 75(70)-9, 71(52)-45, 66-25, 76-34, 85(54)-0.
Laxman Rawat (Delhi) bt Manan Chandra (BPCL) 68-55, 122(122)-8, 63-2, 75(74)-14, 64-42; Andy Lee (Hong Kong) bt Faisal Khan (Rlys) 20-69, 6-76, 80-25, 70-19, 62-58, 31-84, 68-14, 62-17.
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