Korda threatened to run away with the second set when he won three games in a row and earned a chance for a double-break lead. Alcaraz held on until it came time for Korda to serve for the match.
Keeping the pressure on, Alcaraz secured a love-break and reeled off five straight games to force a final set.
Undaunted, Korda snapped the streak to get back on the board and weathered some stellar play from Alcaraz to score the first break of the decider.
Down 3-5 for a third straight set, Alcaraz made another brave last stand to force Korda to serve for the match. Korda made no mistake this time, putting away a backhand to set up two match points. Alcaraz saved the first but Korda claimed the second with one last well-struck serve.
Standing between Korda and a return trip to the quarterfinals will be either No. 14 seed Karen Khachanov or Spanish qualifier Martín Landaluce.
