Rubio told CNN that "as our intelligence agencies have said, these leaks are an effort by a foreign government to interfere with our electoral process, and I will not indulge it."
His remarks were first reported by ABC News.
"Further, I want to warn my fellow Republicans who may want to capitalize politically on these leaks: Today it is the Democrats. Tomorrow it could be us," he told ABC.
The Florida senator -- engaged in a tough reelection fight against Democratic Rep. Patrick Murphy -- said that he "will not discuss any issue that has become public solely on the basis of WikiLeaks."
Rubio's position is a departure from the attacks launched by Republicans -- led by presidential nominee Donald Trump and his running mate Mike Pence -- who have seized on the emails to portray Clinton as politically calculating and dishonest with voters.
One of the hacked emails from campaign chairman John Podesta contained apparent transcripts of Clinton's private speeches, where she advocates for "open borders and free trade," which her campaign later downplayed as a generalization of her views on trade.
Trump, though, seized on the email to press his case against illegal immigration.
"Hillary Clinton's radical call for open borders ... would end the United States as we know it," Trump said at a Pennsylvania rally this month. "Because in the WikiLeaks, it was all about open borders, free trade for everybody."
Rubio's argument is actually much closer to the talking points pressed by the Clinton campaign itself, whose aides and representatives have said the revelations contained in the emails should be discounted due to the illegal means by which the emails were obtained.
This month, shortly after WikiLeaks began publishing emails from the Clinton campaign, Clinton spokesperson Brian Fallon tweeted, "@wikileaks You are no media organization. You are a propaganda arm of the Russian government, running interference for their pet candidate, Trump."
"Media needs to stop treating WikiLeaks like it is same as FOIA. Assange is colluding with Russian government to help Trump," he also wrote.
US officials told CNN
that there is mounting evidence that the Russian government is supplying WikiLeaks with the hacked emails. The methods of the disclosures "suggest Moscow is at least providing the information or is possibly directly responsible for the leaks," one US official said.