The Russian world number 13 won 7-5, 6-4 and will face the talented American teenager Iva Jovic in the quarter-finals in eastern China.
The 21-year-old Eala in August became the first player from the Philippines to win a WTA title and is typically followed by a throng of fans from her country.
But after beating France's Diane Parry in straight sets in the previous round in Ningbo, the world number 20 came up against a far sterner test in Shnaider.
The 22-year-old Russian, who is seeking to win a sixth WTA singles title, used her superior power to take the first set in just over 50 minutes.
The second set was closer as Eala fought for survival.
Shnaider put the Eala serve under severe pressure in the ninth game before breaking for 5-4, then doggedly held to seal victory in one hour and 55 minutes.
The 18-year-old Jovic defeated two-time Grand Slam finalist Jasmine Paolini of Italy 6-4, 6-3.