The vote came after weeks of talks between the lawyers for the committee and the attorney general failed to yield an agreement over access to Mueller's full, unredacted report.
White House Press Secretary Sarah Sanders said the privilege claim was in response to Nadler's "blatant abuse of power" and "at the attorney general's request".
Those dozen lawmakers include the top Republicans and Democrats on the House and Senate Judiciary and Intelligence Committees, as well as the House and Senate leaders from both parties.
Representative Jerrold Nadler, the panel's chairman, said Barr's refusal to comply with the requests of U.S. lawmakers amounted to a "constitutional crisis".
"The Attorney General has been transparent and accommodating throughout this process, including by releasing the no-collusion, no-conspiracy, no-obstruction Mueller Report to the public and offering to testify before the Committee". The Department of Justice tends to reject prosecuting criminal contempt cases when executive privilege is invoked.
The White House assertion of privilege represents the latest collision between Trump and House Democrats, who have seen their investigations of the president blocked at every turn.
While Pelosi still opposes introducing articles of impeachment against Trump, Pelosi said Wednesday that Trump is "becoming self-impeachable" due to his efforts to ignore Congressional subpoenas and prevent key individuals from testifying before Congress. Pelosi added that Barr, who last week also refused to testify before the House panel, should be held in contempt of Congress.
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However, the House would still need to send a criminal referral to the United States attorney for the District of Columbia, a Justice Department official who is likely to defend the Attorney-General. "They seemed to be hoping for a national crisis", McConnell said on the Senate floor Tuesday. The top Republican on the Judiciary Committee, Rep. Doug Collins of Georgia, said "Democrats have launched a proxy war smearing the attorney general" when their anger actually lies with the president.
If the committee holds Barr in contempt, it would be the first step in what could be a protracted, multipronged court battle between Congress and the Trump administration.
But Mr Mueller detailed 10 instances where Mr Trump possibly attempted to impede the investigation. "The Trump Administration's invocation to protect the special counsel report in the wake of a subpoena from the House Judiciary Committee is, however, odd, at least from a legal, nonpolitical standpoint". He derided what he said was the "outrage industrial complex" of Democrats and television news show pundits over special counsel Robert Mueller's conclusion that Trump did not collude with Russians to help him win.
The measure may be largely symbolic as few expect the Department of Justice to indict its own head with criminal contempt.
The Justice Department did not respond to a request for comment on Tuesday's negotiations with Mr Nadler's committee.
Earlier this month, Barr told lawmakers that the Department of Justice has "multiple criminal leak investigations" underway in response to unauthorized media contact by department officials during Mueller's Russian Federation investigation.