The senators want the Iraqi government to handle security
The Democratic-controlled Senate Foreign Relations Committee dismissed Mr Bush's policy as "not in the national interest" in a 12-9 vote.
It comes a day after the president gave his State of the Union address asking Congress to give it "a chance to work".
Both votes are non-binding but may put pressure on Mr Bush to reconsider.
The resolution opposes Mr Bush's plan to send 21,500 additional troops to Iraq, the majority of them to violence-hit Baghdad in an effort to improve security and end sectarian clashes.
President Bush has made his decision on Iraq; the fact of the matter is we need to get the job done
Vice President Dick Cheney
An advance guard of 3,200 troops arrived in Baghdad on Sunday.
The BBC's Justin Webb in Washington says if the resolution is supported by key senators from both sides of the aisle, the rebuff suffered by the White House will be all the more visible and difficult to ignore.
'The grinder'
The resolution was proposed earlier this month by three leading senators who said the plan was not in US interests, urging an early transfer of security to Iraqi leaders.
The three senators were Democrats Joseph Biden and Carl Levin, and Republican Chuck Hagel, a long-standing critic of the war.
"We better be damn sure we know what we're doing, all of us, before we put 22,000 more Americans into that grinder," said Senator Hagel, the only Republican to support the resolution.
Senator Biden, the panel's chairman, said the legislation was "not an attempt to embarrass the president".
"It's an attempt to save the president from making a significant mistake with regard to our policy in Iraq," he said.
'In disarray'
Before the resolution goes to the full Senate, Mr Biden has said he is willing to negotiate changes in hope of attracting support from more Republicans.
A growing list of Republican senators - including Gordon Smith, George Voinovich and Sam Brownback - have said they will back the proposals, AP news agency reports.
However, Republican Richard Lugar said the resolution would force nothing on Mr Bush but would confirm to US friends and allies that it was "divided and in disarray".
The response from Vice-President Dick Cheney was defiant.
"President Bush has made his decision on Iraq; the fact of the matter is we need to get the job done," he said.
Mr Cheney described suggestions that the administration's credibility was on the line as "hogwash".
Democrats in the House of Representatives intend to hold a vote shortly after the Senate.
Kontrolowana przez demokratów Komisja Senacka zajmująca się relacjami z zagranicą odrzuciła politykę prezydenta George'a Busha jako niezgodną z narodowym interesem. Jego plan związany jest z wysłaniem dodatkowych 21 500 żołnierzy do Iraku, w przeważającej części do Bagdadu (pierwsze 3200 z tej liczby przybyło już do tego miasta w niedzielę). Ta uchwalona rezolucja większością 3 głosów (12:9) nie jest wiążąca, ale - jak ogólnie rzecz biorąc wynika to ze słów przewodniczącego Komisji - jest swego rodzaju wskazówką dla Prezydenta sugerującą mu ponowne przemyślenie swoich planów i punktem odniesienia do tego, co może się dziać podczas głosowania nad tą sprawą tym razem przez cały Senat w następnym tygodniu.