The 26th seed finally sealed the quarter-final on his sixth match point and plays Poland's 16th seed Hubert Hurkacz for a place in Sunday's final.
With the centre-court roof closed because of rain in Shanghai, the first set went to a tiebreak after they exchanged breaks in the 11th and 12th games.
US Open semi-finalist Shelton, who uncorked a serve clocked at 236 kilometres (150 miles) per hour earlier in the set, had Korda on the back foot throughout a gripping tiebreak.
The pressure finally told when Shelton unleashed an ace on his fifth set point that Korda could only watch whizz by.
Korda, whose father Petr reached two in the world at his peak, recovered to break Shelton's big serve at the start of the second set and raced to level the quarter-final at one set all.
Korda broke the 19th seed at the start of the deciding set and then dug himself out of a hole, saving three break points to extend his lead to 3-1 and then 4-2.
But Korda, whose only previous ATP title came in 2021 in Parma, allowed Shelton back in again and they went to another tiebreak.
Korda let slip five match points from 6-1 in the tiebreak before belatedly sealing an absorbing encounter in just under three hours.
In the first quarter-final, Hurkacz similarly fought back to defeat unseeded Hungarian Fabian Marozsan 4-6, 6-1, 6-3.
The Shanghai Masters is back after four years following the pandemic and has witnessed a succession of shocks.
Top seed Carlos Alcaraz was turfed out in the last 16 to follow Daniil Medvedev - beaten by Korda - Holger Rune, Stefanos Tsitsipas and Jannik Sinner out the exit door.
There is only one player from the world's top 10 left in the draw in the form of seventh-ranked Russian Andrey Rublev.
The fifth seed faces 32nd seed Ugo Humbert of France in the quarter-finals while 18th seed Grigor Dimitrov - who turfed out Alcaraz - plays 22nd seed Nicolas Jarry.
World number one Novak Djokovic skipped the tournament.
