Democrats have been critical of Trump's choice because of the former ExxonMobil head's alleged close ties to Russia.
Tillerson received full support from the Republicans in the Senate and backing from four Democrats.
Betsy DeVos is looking to be confirmed as education secretary in the coming days, and White House press secretary Sean Spicer said Wednesday he was "100%" confident that will happen.
But Republican Sens. Susan Collins of Maine and Lisa Murkowski of Alaska are speaking out against DeVos, with both senators
saying Wednesday they planned to vote against her -- the first Republicans to speak out against the nominee.
Now, the White House and congressional leadership need 50 votes for DeVos and have Vice President Mike Pence break the tie in the final tally.
Ten Democratic senators boycotted
-- and in turn delayed -- the vote for Oklahoma attorney general Scott Pruitt as head of the Environmental Protection Agency.
"If Scott Pruitt is serious about serving as our next EPA Administrator, he should be more than willing to provde @EPWDems complete answers," Democratic Sen. Tom Carper tweeted.
But Republicans argued the move was not about the nominee and position, but instead a jab of partisan politics intended to slow the confirmation process.
"Not having a vote on this nominee today, not organizing this important committee, is a shame," Wyoming Republican Sen. John Barrasso, chairman of the committee, said before adjourning.
"Temper tantrums waste a lot of energy but they don't accomplish anything," Sen. Dan Sullivan of Alaska said.
Republicans on the Senate Finance Committee, tasked with confirming two Cabinet nominees, moved ahead without their Democratic counterparts.
For the second day in a row, Senate Democrats didn't show up to the meeting. Committee Chairman Sen. Orrin Hatch called it "extraordinary circumstances" and gave room for the Senate GOP in attendance to vote on suspending the rules of the committee.
The 14 Republicans voted to move both Steve Mnuchin and Rep. Tom Price, the respective nominees to lead the Treasury and Health and Human Services departments, to the full Senate for a vote.
"Today, for the first time in history, the Senate Finance Committee broke the rules to push through on a partisan basis two nominees," Sen. Ron Wyden, D-Oregon, said in a statement. "Congressman Tom Price, whose stock trades call into question whether he will work in the public interest or his own, and the other, Steven Mnuchin, who appears to have misled the committee on his company's foreclosure practices after the Great Recession."
Alabama Sen. Jeff Session is one step closer to becoming US attorney general after the Senate Judiciary Committee voted to advance
his nomination following a procedural delay from Senate Democrats.
Eleven Republicans voted for Sessions, while nine Democrats voted against him.
The vote was initially planned for Tuesday, but on Wednesday, senators continued to voice their opposition.
"We have an important job to do here and it's important that we understand the nominee's record accurately -- it's not our job to shade the record," Sen. Al Franken, D-Minnesota, said about Texas Republican Sen. Ted Cruz's representation of Sessions from a previous hearing.
The vote to confirm Sessions could be as early as next week.
Trump has advised Republicans to "go nuclear" if Democrats attempt to filibuster his Supreme Court nomination of Neil Gorsuch.
Gorusch met with Pence and Republican leaders on Capitol Hill Wednesday morning.
"If we end up with that gridlock, I would say, 'If you can, Mitch, go nuclear,' " Trump said of Senate Majority Leader Mitch McConnell. "That would be an absolute shame if a man of this quality was caught up in the web."
The "nuclear option" would change the process of confirming Supreme Court nominees down the road, and involves removing the requirement for 60 votes to break a filibuster.
Trump nominated Gorsuch to replace late Justice Antonin Scalia on Tuesday night, and Senate Judiciary Committee Chairman Chuck Grassley told CNN he plans to have the confirmation hearing in six weeks.