The third seed lost 6-7 (9-11), 2-6 to the 29th-seeded Eala in a match lasting two hours and 15 minutes.
The first set lasted 85 minutes, with Świątek having two set points of her own.
But she struggled in the second set, as unforced errors, serving trouble and mounting frustration took hold.
It was the pair's third career meeting, and Świątek's second defeat to the Filipino player.
The early exit means the 25-year-old Polish tennis star is set to drop from third to at least sixth in the world rankings.
Świątek was also eliminated in the third round of the 2024 Wimbledon Championships before going on to win the title in 2025.
Eala will next face Italy's Jasmine Paolini, who beat Greece's Maria Sakkari 6-1, 6-2.
In the singles, Hubert Hurkacz remains the only Pole left in the draw.
He reached the last 16 on Friday with a 4-6, 7-6 (7-5), 7-5, 6-2 win over American Tommy Paul.
He will next play Germany's Jan-Lennard Struff, who beat Russia's Daniil Medvedev, for a place in the quarter-finals on Sunday.
(ał)
Source: IAR, PAP