WEST PALM BEACH, Fla.—Democratic front-runner Hillary Clinton won critical presidential primaries in the industrial Midwest and completed her sweep of the South on Tuesday, as she took full command of a contest that has been far more competitive than expected at the start.

Rival Bernie Sanders was still hoping to win in Missouri, where the result was too close to call. But even a Missouri win for Mr. Sanders seemed nowhere near enough to change the dynamic of the race. Mrs. Clinton won in Ohio, Illinois, Florida and North...