HOUSTON — Ted Cruz, an insurgent backed by the Tea Party, easily defeated the candidate favored by Gov. Rick Perry on Tuesday in a runoff election for the Republican Senate nomination that revealed a wide rift in Texas between the party establishment and restless anti-incumbent activists on the right.

With the come-from-behind victory over Lt. Gov. David Dewhurst, Mr. Cruz is heavily favored to win the Senate seat being vacated in November by Kay Bailey Hutchison and appears likely to become a star of the national conservative movement. On Tuesday night, his unofficial margin of victory was 57 percent to 43 percent.

“We did it,” Mr. Cruz told the cheering crowd gathered at a Marriott Hotel in Houston to claim victory. “Millions of Texans, millions of Americans are rising up to reclaim our country, defend liberty and restore the Constitution.”

Mr. Cruz, 41, is the latest conservative rebel to bring down an established party leader, tapping into frustration within the Republican ranks nationwide. These dissident triumphs include, in this year’s primaries, the defeat of Senator Richard G. Lugar of Indiana by Richard E. Mourdock and Deb Fischer’s win over a veteran Republican for the Senate nomination in Nebraska. They also echo Marco Rubio’s Senate victory in 2010 over a Republican governor, Charlie Crist of Florida.