The 2022 Wimbledon champion and reigning Australian Open winner burst out of the blocks to take a 4-1 lead but Eala fought back, breaking the Kazakh's usually strong serve twice to take the opening set.
World number 35 Eala stayed in control during the second, breaking her heavily favoured opponent once to storm to an upset victory.
The 21-year-old Eala exacted revenge over Rybakina, who beat her in straight sets on clay at the Italian Open in May.
Eala, the first Filipino player to break into the top 50, is in fine form, having beaten newly minted Queen's Club champion Donna Vekic in the opening match in Berlin.
Eala will face world number eight Elina Svitolina, who beat German Eva Lys in straight sets 6-3, 6-2 earlier on Thursday, in the quarter-finals.
In the other quarter-finals, world number one Aryna Sabalenka takes on Nikola Bartunkova, world number four Jessica Pegula faces Madison Keys and Linda Noskova is up against Paula Badosa.