Both Corker and Cardin made their opening salvos in support of the legislation.
A Cruz spokeswoman suggested a procedural vote was "the only vote that mattered."
The vote came more than 160 days after she was nominated by President Obama.
But Majority Leader Mitch McConnell (R-Ky.) said there will be debate only this week.
In a 66 to 34 vote, senators moved toward confirming her as the next attorney general.
The Senate leader said the bill will be "open for amendment."
The Senate could take a final vote on Loretta Lynch's nomination Thursday.
But senators rejected an amendment she was pressing for them to adopt.
Paul gave a fist-bump to Harry Reid on the Senate floor.
Obama's long-delayed nominee for attorney general could get a final vote as soon as Thursday.