Murray, 35, was given a wildcard for the tournament - his first since September's Laver Cup.
Having lost the opener, Murray dominated the second set before seeing Korda do the same to wrap up a 6-4 1-6 6-1 win in two hours and 27 minutes.
The Briton was broken by his American opponent in the third and seventh games of set one to fall 5-2 behind, subsequently breaking back in the eighth but unable to turn things around.
Murray then took the first five games of set two en route to drawing level - only to lose the first five of set three as Korda surged to victory.
Korda will now face Frenchman Arthur Rinderknech, who beat Spain's second-seeded Pablo Carreno Busta (Spa) 4-6 6-3 7-6 (16).
Korda moves on to a semi-final showdown against France's Arthur Rinderknech, with either Dominic Thiem or top seed Andrey Rublev set to meet the victor of that contest in the final.
The Gijon Open is one of the six tournaments granted a one-year license for this season after the China leg of the ATP Tour was cancelled because of ongoing Covid-19 restrictions.
Murray has yet to confirm if he will play again this year or opt to rest before beginning his build-up towards the Australian Open, which starts in Melbourne on January 16.