Cruz, 41, a former state solicitor general, advances to the general election in November, when Texas voters will elect a successor to the retiring Republican Sen. Kay Bailey Hutchison. He is favored to win and seems headed for prominence among U.S. conservatives, The New York Times said.
Former Texas state Rep. Paul Sadler defeated teacher Grady Yarbrough in Democratic voting Tuesday.
During the current election cycle, insurgents have taken down several establishment conservative Republicans, including longtime Sen. Richard Lugar of Indiana, who lost a primary campaign against Richard E. Mourdock, the newspaper noted.
Dewhurst, 66, the Texas lieutenant governor and a longtime ally of Republican Gov. Rick Perry, had been the front-runner in the Senate campaign.
Since finishing second in Texas' May 29 primary, Cruz -- with funding from Tea Party groups and endorsements of 2008 GOP vice presidential candidate Sarah Palin and conservative Sen. Jim DeMint, R-S.C. -- has caught the voters' imagination and many polls showed him pulling ahead.
In other Texas races, handicappers said the Democratic race in the 23rd Congressional District, where former Rep. Ciro Rodriguez was dueling state Rep. Pete Gallego. Roll Call said this district is likely the Democrats' best and only opportunity to flip a congressional seat in Texas this year. The district is represented by freshman Rep. Francisco Canseco.
Texans in the 14th Congressional District will choose a successor to retiring Rep. Ron Paul, the libertarian-leaning representative who stirred the pot when he sought the Republican presidential nomination this year. Roll Call said the GOP contest is a tossup, with Texans divided between state Rep. Randy Weber and Pearland City Council member Felicia Harris.
The winner will face former Rep. Nick Lampson, the Democrat who once represented the Beaumont area of the district.
Voters also went to the polls Tuesday in Georgia, where observers said the Republican races in the 9th and 12th Congressional Districts are the ones to watch, Roll Call said.
While the winner of the reliably Republican 9th district would be virtually assured of joining Congress, the 12th Congressional District GOP winner must best Rep. John Barrow, who is considered vulnerable, Roll Call said.
Also, four Republicans are running viable campaigns in the 12th Congressional District, making an Aug. 21 runoff likely, the Washington publication said.
Notable deaths of 2014 [PHOTOS]