Hillary Clinton won sweeping primary victories in Texas and throughout the South on Super Tuesday, a sign that the Democratic presidential front-runner was vaulting to a commanding lead in the race for the nomination and leaving rival Bernie Sanders diminishing space to stage a comeback.

Mr. Sanders captured his home state of Vermont and Oklahoma. But Mrs. Clinton easily won the delegate-rich Southern states of Virginia, Georgia, Alabama, Tennessee and Arkansas, as well as Texas, the Associated Press projected—propelled by...