Brazile resigned from CNN as a contributor on October 14. Her work there had been suspended when she became interim chairman of the Democratic National Committee in July, a CNN spokeswoman said.
She became head of the party organisation after chairman Debbie Wasserman Schultz was forced to resign by the release of e-mails hacked from the DNC that showed favouritism for Clinton over Sanders during the primaries. US intelligence agencies have concluded those communications were stolen and released by agents of the Russian Government in an effort to interfere in the US elections.
Left unresolved is how Brazile would have received advanced copies of the debate questions that were supposed to be closely guarded, if the e-mails are genuine.
Lauren Pratapas, a CNN spokeswoman, said that the cable news network "never gave Brazile access to any questions, prep material, attendee list, background information or meetings in advance of a town hall or debate".
"We are completely uncomfortable with what we have learned about her interactions with the Clinton campaign while she was a CNN contributor," Pratapas said.
White House press secretary Josh Earnest told reporters that President Barack Obama considered Brazile a "true professional".
"The President believes that she's done a fine job stepping in in a very difficult position" to lead the Democratic Party, Earnest said.
The question on the death penalty was written by that debate's co-moderator, Roland Martin of the TV One network. Politico quoted Martin as saying he didn't "believe" he had shared it with Brazile. Neither TV One nor Martin were involved in the earlier debate, which was the subject of the alleged e-mail leaked today.
- Bloomberg