NEW YORK—The euro sank to the weakest point in more than four years against the dollar as a disappointing U.S. jobs report, combined with fears the euro-zone sovereign-debt crisis is spreading across the region, led investors to flee risky assets.

The common currency fell below $1.20—the 10-year average—and is on a course that could bring it to $1.18, where it exited the first day of trading when the euro was introduced in 1999.

The...