Nadal, seeking his first Miami title, had little trouble in his 6-2, 6-2 victory over the 25th-seeded Radek Stepanek, remaining perfect at 6-0 in his career against the Czech veteran.
The Spanish star, who was bested by Novak Djokovic in last year's final, now goes on to face 16th-seeded Kei Nishikori of Japan, who moved on with a 6-4, 6-2 verdict over unseeded Czech Lukas Rosol.
Murray, meanwhile, advanced to the fourth round after Canadian opponent Milos Raonic withdrew with an ankle sprain.
Sixth-seeded Jo-Wilfried Tsonga of France also moved on with a 6-4, 7-6 (7-2) win over 32nd-seeded German Phillipp Kohlschreiber. Next up for him is 19th-seeded Florian Mayer of Germany, who got past No. 10 seed John Isner of the United States 6-4, 6-2.
In another upset, seventh-seeded Czech Tomas Berdych fell 6-3, 2-6, 6-4 to Belgian Grigor Dimitrov.