World number six Rune came back from a set and a break down to defeat Grigor Dimitrov 3-6, 7-6 (8/6), 7-6 (7/4), 6-3.
He will tackle Alcaraz in the last eight if the top seed gets past former finalist Matteo Berrettini in his fourth-round clash.
"They are both great players, but I am not afraid. I am excited," said Rune, who was supported on Court One by Denmark's Crown Prince Frederik.
Rune, already a two-time quarter-finalist at the French Open, had been just two points from defeat in the previous round before seeing off Alejandro Davidovich Fokina in five sets.
On Monday, Dimitrov, the 24th-ranked Bulgarian who made the semi-finals in 2014, pocketed the first set after carving out a key break in the eighth game.
![Denmark's Holger Rune (R) shakes hands after winning against Bulgaria's Grigor Dimitrov Denmark's Holger Rune (R) shakes hands after winning against Bulgaria's Grigor Dimitrov](https://livesport-ott-images.ssl.cdn.cra.cz/r900xfq60/2d2ae2f5-d9c4-4922-a22a-c837f00e5d44.jpg)
The veteran was a break ahead early in the second set, which 20-year-old Rune retrieved in the eighth game.
He levelled the match in the tiebreak.
After securing the third set, also on a tiebreak, Rune broke for 4-2 in the fourth courtesy of back-to-back double faults by Dimitrov.
Rune wrapped up the match, becoming the first Dane since Kurt Nielsen in 1958 to make the quarter-finals when Dimitrov pushed a weary backhand wide.
"Grigor is an amazing player, he pushed me to the limit," said Rune after a performance in which he fired 11 aces and 46 winners in total.
"It was a crazy match, tough and long. But I always fight to the end.
"I kept telling myself that Wimbledon is only once a year and that I have to fight to have any chance of winning."