With 51 percent of precincts reporting, Udall had an 11-point lead over Schaffer, 53 percent to 43 percent, CNN reported.
Udall's win boosted the Democratic majority in the Senate. The seat is currently held by Sen. Wayne Allard, a Republican, who is retiring.
Udall comes from a political family. His cousin, Tom Udall, will serve alongside him in the Senate, representing New Mexico.
Udall's father was the late Arizona Rep. Morris "Mo" Udall of Arizona, and his uncle was former Interior Secretary Stuart Udall, who served under Presidents John Kennedy and Lyndon Johnson.