World number two and top overall seed Alcaraz was strong in his 6-4 7-6 (7-0) victory over Serbia's Dusan Lajovic, improving his unbeaten record in 2023 to 6-0.
Alcaraz has his eyes set on his second title win in a row after also emerging triumphant in last week's Argentina Open.
Meanwhile, Norrie had to fight back from an early deficit to eliminate Hugo Delien 4-6 6-1 6-4, with the world number 13 and second seed now boasting eight wins from his past nine matches. The only loss during that stretch was to Alcaraz in the Argentina Open decider.
For a chance to avenge that loss, Norrie will have to beat Bernabe Zapata Miralles after he overcame Albert Ramos-Vinolas 6-4 2-6 6-4 in an all-Spanish quarter-final.
Alcaraz will contend with Nicolas Jarry, after the unseeded Chilean knocked out his second seeded opponent of the tournament, this time getting the better of Sebastian Baez 6-3 7-6 (7-3).
With Norrie having already set up a quarter-final against last year's runner-up Sebastian Korda, the Briton was joined in the last eight by the second and third seeds.
Grigor Dimitrov defeated Mitchell Krueger to mark his first appearance since a disappointing second-round exit at the Australian Open in style.
The Bulgarian won 7-6 (7-4) 6-3, with Serena and Venus Williams watching on from the crowd in far from ideal conditions in Florida.
"It's always very tough [in your opening match], especially playing the last match on," Dimitrov said.
"I've been here for a week already and I was eager to get out there and play. Today was just a match I had to get through. Very tough conditions: rain, wind, cold, hot. It is what it is, and you've got to deal with it."
Opelka, who won the tournament in 2020, went the distance against fellow American Jack Sock, eventually prevailing 6-7 (6-8) 7-6 (7-4) 6-1.
He will face Adrian Mannarino in the last eight, after the Frenchman's 6-4 2-6 7-5 defeat of Brandon Nakashima. Dimitrov will play Australian John Millman.
At the Rio Open, only two of the four scheduled matches were completed due to rain.
Qualifier Miomir Kecmanovic dispatched sixth seed Lorenzo Sonego 7-5 6-4, while Francisco Cerundolo overcame Roberto Carballes Baena.
Top seed Matteo Berrettini's clash with Brazilian Thiago Monteiro was pushed back to Friday, while Carlos Alcaraz will resume his match against Federico Delbonis in a strong position in the first set.
Verdasco, the ninth seed at the ATP 500 tournament, was beaten 6-3 6-3 by fellow Spaniard Pablo Andujar on the clay in Brazil.
The 36-year-old has fallen to three straight losses, the first of which was a third-round defeat at the Australian Open.
Verdasco – the 2018 runner-up – was one of three seeds tested, but Garin survived a test against Andrej Martin.
Garin, who won the title in Cordoba earlier this month, needed more than three hours to edge Martin 4-6 7-5 7-6 (7-5).
Albert Ramos-Vinolas departed, the seventh seed falling to 16-year-old Carlos Alcaraz 7-6 (7-2) 4-6 7-6 (7-2) after three hours, 36 minutes.
Federico Delbonis advanced, while wildcard Thiago Seyboth Wild needed almost four hours to get past Alejandro Davidovich Fokina 5-7 7-6 (7-3) 7-5.
There were also upsets at the Delray Beach Open, where John Millman and Adrian Mannarino were beaten in the first round.
Millman, the fifth seed, went down to Yoshihito Nishioka 3-6 6-4 6-2 and seventh seed Mannarino fell to Kwon Soon-woo 1-6 6-3 6-2.
Frances Tiafoe, the champion in 2018, came through his opener against qualifier Emilio Gomez.
