Cornyn defeated Rep. Steve Stockman's challenge by more than 40 percentage points, while Pete Sessions walloped local Tea Party leader Katrina Pierson by a two-to-one margin, the Hill reported Wednesday.
Rep. Ralph Hall, Congress's oldest member at age 90 and a 17-term incumbent, was forced into a runoff race by former U.S. Attorney John Ratcliffe, who ran to Hall's right. A candidate must get more than half of the total vote to win a party's nomination.
In another contested GOP primary, Houston lawmaker and radio host Dan Patrick rode the anti-incumbency wave to top incumbent David Dewhurst in the lieutenant governor's race Tuesday, even though the two will meet in a May 27 runoff, the Dallas Morning News reported.
In the party primaries for gubernatorial candidates, voters propelled Republican Attorney General Greg Abbott, a Republican, and state Sen. Wendy Davis of Fort Worth, a Democrat, toward their showdown to succeed Gov. Rick Perry, who is stepping down after 14 years.