The 23-year-old followed up his victory over 12-time grand slam champion Novak Djokovic in the previous round by sweeping past last-16 opponent David Goffin.
Edmund’s 6-3, 6-3 win booked a last-eight clash against Canadian Denis Shapovalov, who beat Milos Raonic 6-4, 6-4.
It also means that the British No 1 will secure a place in the world’s top 20, and he beat Shapovalov when they last met in Brisbane earlier this year. “There were lots of good things. I managed my game very well today,” Edmund told Sky Sports.
“My game is one of those where the better I manage it and pick my moments to be aggressive, then I play well.
“I served really well today. It got me through easy games and also got me out of some tricky situations.
“He is a top player, top ten in the world, so it is very pleasing.”
Edmund built on the Djokovic display in emphatic fashion, taking control from the opening game, when he broke world No 10 Goffin’s serve.
He then kept the Belgian on the back foot throughout and forced several errors from his opponent.
After breaking Goffin’s serve in the opening game, Edmund went on to lose only four points in his first four service games and took the set in 32 minutes.
Edmund saved two break points on his first service game in set two, digging deep as Goffin tried to force his way back into the contest.
But he then closed it out by clinching victory with a third match point, sealing a last-eight spot in one hour and 12 minutes.
Rafael Nadal broke John McEnroe’s record for consecutive set wins on a single surface as he also moved into the quarter-finals. Nadal beat Argentina’s Diego Schwartzman 6-3, 6-4 on the clay to take his winning set streak to 50 and eclipse McEnroe’s total which the American achieved on carpet in 1984.
Nadal, who is looking for a sixth career Madrid Open title, broke Schwartzman in the sixth game of each set and appeared to be on course for a comfortable win.
World No 16 Schwartzman served up a minor scare by breaking back for 4-4 in the second set, but Nadal responded immediately and closed out for a history-making win.
Nadal will next face the man he beat in last year’s final, Dominic Thiem, after the fifth seed came from a set and a break down to edge out Croatian Borna Coric 2-6, 7-6 (5), 6-4.
Previously in-form fourth seed Juan Martin Del Potro suffered a shock 3-6, 6-4, 7-6 (6) defeat by Dusan Lajovic, handing the Serbian the first win over a top-ten player of his career.
Lajovic will next face South Africa’s Kevin Anderson, who reached his first Madrid quarter-final with a 6-3, 7-6 (7) triumph over Philipp Kohlschreiber of Germany.
The final quarter-final will be between second seed Alexander Zverev – a 6-4, 6-2 winner over Leonardo Mayer, and John Isner, who edged Pablo Cuevas 6-7 (9), 7-6 (3), 7-6 (4).