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 No.2 and No,3 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 because of rain.
Qualifier Miomir Kecmanovic dispatched No.6 seed Lorenzo Sonego 7-5 6-4, while Francisco Cerundolo overcame Roberto Carballes Baena.
No.1 seed Matteo Berrettini's clash with Brazilian Thiago Monteiro was pushed back to Saturday (AEDT), while Carlos Alcaraz will resume his match against Federico Delbonis in a strong position in the first set.