The Italian world No.1 had earlier brushed aside Britain's Cameron Norrie 6-2 7-5, stretching his ATP Masters 1000 winning streak to 25 matches.
Jodar, 19, followed up his three-set victory over 27th seed Joao Fonseca with another assured display against world No.66 Kopriva under Madrid's fierce mid-afternoon sun.
Tall and lean, Jodar traded heavy groundstrokes from the baseline with his quick-footed opponent in a tightly contested opening set. At 6-5 to Jodar and a tiebreak looming, the Spaniard raised the intensity, breaking decisively to seal the set, drawing roars from the stands.
The second set was one-way traffic as the clay specialist booked a mouth-watering meeting with Sinner on Wednesday (local time).
The Italian is chasing a fifth consecutive Masters 1000 title and his first in Madrid.
Jodar has enjoyed a meteoric rise, climbing from outside the top 900 to world No.42 in a year, winning his first ATP title in Marrakech this month, and establishing himself as one of tennis's most promising young talents.
"Every match in a tournament like this is tough and can get complicated, so you always have to keep the same mindset. That's what I did in the first set: I dug deep, played my best tennis, and I'm really happy with the win," Jodar said.
Sinner raced through his opening set in 35 minutes, giving Norrie scant opportunity to settle.
In the second the Briton, seeded 19th, went 3-2 down after an underarm serve trickled into the net, drawing boos from the crowd, but immediately broke back. However, from 5-5 Sinner broke again before serving out.
Sinner subsequently criticised the scheduling of the two-match night sessions, which begin at 8pm local time. Jodar's match with Fonseca had finished just before 1am on Monday.
"For our body and mind, going on court at 11pm and you play at midnight, past midnight, it's not easy," Sinner said.
"Then also, even if you have a day off, the fans they see only us on court, and then we finish at 1.15am. But then you have press conference, recovery, eating, treatment, you don't go to bed until 4 or 5am. It messes up the whole day. We can do better, definitely."
Defending champion Casper Ruud looked to be heading for the exit against a resurgent Stefanos Tsitsipas, but saved two match points in the deciding set before claiming a 6-7 (4-7) 7-6 (7-2) 7-6 (7-3) victory.
Also through to the last eight is in-form young Frenchman Arthur Fils, one of the players who will look to challenge Sinner in the absence of Carlos Alcaraz, with the 21st seed beating Tomas Martin Etcheverry 6-3 6-4.
Fils next faces Czech Jiri Lehecka, who upset sixth seed Lorenzo Musetti 6-3 6-3.
In later action, Italy's Flavio Cobolli, the 10th seed, upset former world No.1 and seventh seed Daniil Medvedev, of Russia, 6-3 5-7 6-4.
Cobolli's victory set up a clash with second-seeded German Alexander Zverev, who was pushed to three sets before beating Czech 23rd seed Jakub Mensik 6-4 7-7 (4-7) 6-3.
with PA
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