A fast-moving Pacific storm swept across northern California and down the coast to Los Angeles last week, bringing a rare rain delay to Dodger Stadium in the middle of the season opener—and some relief to the vast urban population suffering from the state’s severe drought.

Unlike thirsty cities of the past—such as ancient Carthage in Tunisia, which meticulously captured every drop of its scant rain—metro Los Angeles wasted much of the .36 inch that fell on April 7. Flowing across miles of highways, rooftops and parking lots,...