Beating back a challenge from tea party activists, Sen. Orrin Hatch of Utah easily qualified for the Republican primary ballot in his quest for a seventh Senate term Saturday, but he fell just short of winning the GOP nomination outright.

Mr. Hatch needed support from 60% of delegates at a state GOP convention to dispatch his nearest rival, former state Sen. Dan Liljenquist, and to advance to the nomination without the need for a primary. He got 59% on the second ballot.

As...