Haspel, a 33-year veteran of the CIA, stopped short last week of labeling the agency's past torture program as "immoral", saying it was deemed legal at the time by then-President George W. Bush and the Justice Department.
After Wednesday's closed-door committee vote, the full Senate will likely consider her confirmation next week.
Sen. John Hoeven has said he supports Haspel's nomination.
With the Republican Party holding only a 51-49 margin in the Senate, cut to 50-49 in practice because of McCain's long absence due to terminal brain cancer, the shift of even one senator had the potential to torpedo the nomination.
Amid criticism of Gina Haspel for her role in the Central Intelligence Agency's enhanced interrogation techniques like waterboarding, one green beret made a decision to argue that waterboarding wasn't actually torture.
Arizona Republican Sen. John McCain is urging his colleagues to reject Haspel's nomination.
Haspel played an integral role in the CIA's program, both as a senior leader in the agency's Counterterrorism Center and a supervisor of a secret facility in Thailand.
Warner, of Virginia, is the third Democrat to say he'll vote for Haspel.
Haspel wrote that she has "learned the hard lessons since 9/11".
In a letter sent on Monday to Sen.
Haspel's opponents said they don't believe she adequately explained her role in the destruction of the Central Intelligence Agency videotapes that showed waterboarding.
Despite his own misgivings with President Trump, Brennan has signaled his support for Haspel on Twitter. Rodriguez has said publicly that Haspel knew he meant to take the matter into his own hands and issue the order, because he believed he had the authority to do so.
A career intelligence officer and now acting director, Haspel faces ongoing questions over her work running a covert detention site where terror suspects were brutally interrogated. I'm not one of those people who calls it torture. But by the time he released the statement, two key swing votes on the committee, Manchin and GOP moderate Senator Susan Collins of ME, had already announced they would vote for Haspel.