Fery, who beat fellow Briton Toby Samuel in the first round, will face either seventh seed Francisco Cerundolo or Jenson Brooksby in the last eight.
Former British number one John Lloyd, commentating on Fery’s match for the BBC, was impressed by Fery’s performance.
“I see no reason why he can’t be top 50,” he said.
“He’s quick, nasty to play against, he moves well, isn’t afraid to change it up with the slice, and he likes the big stage.”
Fery started confidently by breaking Mannarino’s serve at the first attempt but the 37-year-old Frenchman fought back from 3-0 down to take the set to a tie-break.
An eye-catching cross-court backhand pass and two superb volleys at full stretch – celebrated with a crowd-energising fist-pump – helped Fery move to set point.
World number 44 Mannarino saved it but Fery kept his nerve to force an error that secured the set.
After a medical timeout to stop his nose bleed, Fery maintained his standards in an entertaining second set. He broke Mannarino in the ninth game with a blistering forehand and then held his serve to complete victory.
Meanwhile, top seed Alex de Minaur of Australia secured his place in the quarter-finals with a straightforward 6-4 6-1 victory over Canada’s Denis Shapovalov.
De Minaur will play American world number 32 Brandon Nakashima in the last eight.