World No.6 Sakkari collected her 24th victory of the year by knocking out Saville, which puts the Greek tied for third-most match-wins on tour in 2022 so far.
Former Top 20 player Saville, on the comeback trail after persistent injuries over the last few seasons, has picked up 12 wins over Top 10 players in her career.
But Sakkari, who had 21 winners to just 10 unforced errors on the day, refused to hand another upset to the Aussie. Sakkari is now into her fourth career tour-level quarterfinal on grass, with her other three grass-court quarterfinal results all coming at Nottingham.
Sakkari earned the first break of the match to take a 3-1 lead after slamming a forehand return winner to close out the game. Sakkari swept through the rest of the opener without facing a break point.
The second set was much closer as Saville, currently on the brink of a Top 100 return, matched Sakkari shot for shot. But a sturdy forehand gave Sakkari the critical break to lead 6-5, and she served out the match with ease in the next game.
Sakkari will be tested by No.6 seed Daria Kasatkina in the quarterfinals. Kasatkina has won all four of their previous matches, including on the clay of Madrid last month. This will be their first meeting on grass.