It was an underwhelming performance from Paul, who hit seven winners compared to a staggering 31 unforced errors.
But it was undoubtedly a scintillating display from Indian Wells champion Draper, who hit 20 winners to just 11 unforced errors.
Seeded fifth in Madrid, Draper is now through to his first clay-court Masters 1000 quarterfinal and is the highest-ranked player left in the draw.
World No. 1 Jannik Sinner and No. 3 Carlos Alcaraz were both absent, with world No. 2 Alexander Zverev, No. 4 Taylor Fritz, and No. 5 Novak Djokovic all having fallen by the wayside.
And, Djokovic's early exit in Madrid has given Draper a huge ATP Rankings opportunity. Heading into the tournament, only 300 points separated world No. 5 Djokovic (4,120) and world No. 6 Jack Draper (3,820), with both men handed a significant chance to earn points in the Spanish capital.
Draper was beaten in the second round a year ago, while Djokovic had no points to defend after choosing to skip the event in 2024.
The Serbian was unable to capitalise, earning just 10 points after falling in straight sets to the unseeded Matteo Arnaldi in his opening match; he will sit on 4,130 once the rankings update post-event.
It has been a different story for Draper, who beat Tallon Griekspoor in round two before progressing through round three after the retirement of 30th seed Matteo Berrettini.
Draper's fourth-round victory over Paul sent him into the quarter-final, and he has so far earned 200 points for reaching the last eight.
That gives him an increase of 170 points on top of the points he had already defended, and he now moves to 3,990 points in the live rankings.
However, things will get even better for him should he win his quarter-final against Arnaldi, who backed up his win over Djokovic with further triumphs over Damir Dzumhur and 16th seed Frances Tiafoe.
Should Draper beat Arnaldi, he would move to 4,190 ranking points - and leapfrog Djokovic in the live rankings.
The Brit would move to 4,790 points should he win the title, hypothetically placing him just shy of world No. 4 Fritz, who will hold 4,815 points post-Madrid.
If Draper defeats Arnaldi and moves to 4,190 points, that would put him in a commanding position to reach a new career-high of world No. 5 after the tournament, though it may not be guaranteed quite yet.
One man who could still top the 23-year-old's point tally is sixth seed Alex de Minaur, who faces 10th seed Lorenzo Musetti on Wednesday night.
De Minaur holds 3,635 points in the ATP Live Rankings and will move to 3,735 if he beats Musetti in their round of 16 contest.
However, the Australian could go on to hold 4,535 ranking points should he ultimately win his first Masters 1000 title this week.
If Draper were to beat Arnaldi, a de Minaur run to the final is the only thing that would deny him a new career-high of world No. 5.
Draper and de Minaur are on the same side of the draw, and could potentially meet in the semifinal.
If de Minaur was to beat Draper in the last four and reach the final, he would hold 4,185 points - enough to overtake Djokovic, but not the Brit unless he triumphed overall.
No other player left in the draw outside of de Minaur can hold more ranking points than Draper's hypothetical semi-final total of 4,190, even if they were to lift the title.