The survey, from Democratic firm Public Policy Polling, gives Capito 50 percent support to Tennant's 36 percent support among West Virginia voters.

Capito also has backing from more than half of independent voters and has significant crossover appeal at this point in the race. She draws 30 percent of Democratic voters away from Tennant. The poll is the first major survey since Tennant announced her candidacy last week. 

A plurality of respondents, 45 percent, sees Capito favorably, while 36 percent have an unfavorable opinion. Tennant is more divisive, with nearly a third of voters seeing her favorably, and a third viewing her unfavorably. 

Tennant may be getting dragged down by President Obama, whose approval rating is significantly under water. Sixty-seven percent of respondents in West Virginia disapprove of Obama's job performance while only 28 percent approve. 

Even a plurality of Democrats, 49 percent, disapproves of the president's job performance.

The West Virginia race is crucial to Republican efforts to win the Senate majority in 2014. The seat is currently held by retiring Sen. Jay RockefellerJohn (Jay) Davison RockefellerSenate GOP rejects Trump’s call to go big on gun legislation Overnight Tech: Trump nominates Dem to FCC | Facebook pulls suspected baseball gunman's pages | Uber board member resigns after sexist comment Trump nominates former FCC Dem for another term MORE (D). 

The West Virginia race is rare in that it's one of few throughout history that will likely pit two female candidates against each other. No woman has ever represented the state in the Senate.  

Capito takes more support than Tennant among both women and men, the poll found. 

PPP also surveyed primary challengers to both Tennant and Capito, but both are still unknown by two-thirds of respondents.

Conservatives have criticized Capito and are still looking for a challenger to the establishment pick. 

Former state Del. Pat McGeehan has entered the GOP primary but he hasn't yet received endorsements from any significant national groups. And the new survey gives Capito strong support among conservatives.

— Updated at 1:20 p.m.