Control of the U.S. Senate was handed to the Democratic Party Tuesday night, as the party flipped key seats in Texas, Kansas, North Carolina and South Carolina, allowing the party to secure control of both chambers of Congress in one of the most closely watched midterm elections in years.

Democrats and Republicans squared up on Wednesday for what could be two years of deadlock in Washington, with the resurgent Democrats vowing to use their resounding midterm election victory to rein in...