The judges had the American southpaw 115-111, 116-110 and 117-109 ahead of his compatriot in the first of four title bouts at the new Barclays Center in Brooklyn.
Alexander, who is now expected face mandatory challenger Kell Brook early next year, was on top from the first bell and opened a cut over Bailey's right eye in the 10th round.
But the 25-year-old's one-sided victory did little to impress a vocal crowd in the first major fight in Brooklyn since 1931.
Both fighters were booed throughout the contest with only Alexander offering any positive steps, and even then he restricted himself to jabs and occasional lefts.
Bailey spent most of the contest trying to get his big right going with little success and the few he landed in the early rounds were not enough to hurt Alexander.
"I know I could have did way better," said Alexander, who improved to 24-1 with the win. "But I had to be cautious, Randall is a veteran. He's got more knockouts than I've got wins."