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More outside cash — about $14.6 million — has flowed into the race than any in the nation this election cycle with the exception of the presidential contest, according to OpenSecrets.org, a nonpartisan website that tracks money in politics.

“The race illustrates that the more extreme conservatives within the Texas Republican Party are on the march, and they’re dominating the discourse,” said James Henson, director of the Texas Politics project at the University of Texas at Austin.

If Mr. Cruz prevails in November, he’ll join a wave of tea-party-backed lawmakers elected to Congress in 2010 that helped Republicans take control of the House and slash the Democrats’ numerical advantage in the Senate. Tea-party-supported winners in the upper chamber that year included Sens. Marco Rubio of Florida, Mike Lee of Utah and Rand Paul of Kentucky.

Another tea-party-backed Senate candidate, Indiana State Treasurer Richard Mourdock, beat longtime Sen. Richard G. Lugar in the state’s primary in May,

Mr. Cruz and Mr. Dewhurst failed to get an outright majority in the state’s Republican primary in May, prompting a runoff. In the contest, Mr. Dewhurst earned 45 percent of the vote to Mr. Cruz’s 34 percent.