Spain's teen phenom won Indian Wells to retake the top ranking last week and kept the momentum going at Hard Rock Stadium as he fired off 26 winners to set up a fourth-round meeting with Australian Open semi-finalist Tommy Paul.
Alcaraz, looking to back up his Indian Wells triumph with a Miami title to complete the coveted "Sunshine Double", barely broke a sweat as Bagnis was simply unable to keep pace with the variety in Alcaraz's game.
Azarenka, who reached the last 16 in New York but lost in her Ostrava opener earlier this month, dropped only three first-serve points in the first set and broke in the seventh game.