In a 14-7 vote, four Republicans joined all the committee's Democrats in favoring the bill -- including Chairman Chuck Grassley of Iowa, Lindsey Graham of South Carolina, Thom Tillis of North Carolina and Jeff Flake of Arizona.
Legislation to protect Mueller, who's leading the Justice Department's Russia investigation, has been batted around in Congress for weeks.
Thursday, Grassley said "transparency and accountability" are in the public interest.
"Because special counsel investigations only occur where there is a conflict of interest within the executive branch, special counsel investigations are usually matters of great national concern," he said in a statement. "And Congress, by exercising its oversight powers, can help the American people to have confidence that these investigations are conducted efficiently and independently."
The vote marked the first time lawmakers advanced legislation to protect Mueller as he investigates potential Russia's interference in the 2016 U.S. presidential election. He is also examining whether any members of Trump's campaign colluded with the Russian government to sway the election result.
Trump has repeatedly called the investigation a "witch hunt" and said he has authority to fire Mueller.
The Special Counsel Independence and Integrity Act -- a combined version of two previously-introduced Senate bills designed to protect Mueller's investigation -- was introduced earlier this month.
Under the bill, Mueller can only be fired for good cause and the reasons must be provided in writing. It also would give any special counsel ten days after a termination to challenge the move in court -- and would preserve staffers, documents and materials of an investigation.
Senate Majority Leader Mitch McConnell has previously said he won't bring the bill to the full Senate for a vote, even if it's approved by the committee.
"I'm the one who decides what we take to the floor ... and we will not be having this on the floor of the Senate," McConnell said last week.
"It's important that we not overstep our constitutional authority," Hatch said.
Experts say if it reached the Senate floor, the bill is unlikely to get the 60 votes needed to pass -- and probably has even less of a chance in the House.