The measure details how the inquiry will move into a more public phase. It was not a vote on whether the president should be removed from office.
This was the first test of support in the Democratic-controlled House for the impeachment process.
The White House condemned the vote, which passed along party lines.
Only two Democrats - representing districts that Mr Trump won handily in 2016 - voted against the resolution, along with all Republicans, for a total count of 232 in favour and 196 against.
The resolution also sets out due process rights for Mr Trump's lawyers under the congressional inquiry.
The Republican president is accused of trying to pressure Ukraine into investigating unsubstantiated corruption claims against his political rival, Joe Biden, and his son who worked with Ukrainian gas company Burisma.
Mr Trump denies wrongdoing.
Republicans have criticised Democrats for the closed hearings - in which Republican lawmakers have also taken part. But Democrats insist they were needed to gather evidence ahead of the public stage of the investigation, and deny allegations they have been secretive.
The resolution moves the inquiry to a new phase, which could eventually see articles of impeachment recommended against Mr Trump. If that happened, and the House voted to pass the articles, a trial would be held in the Senate.
A historic clash ahead
Republicans have been clamouring for weeks for the Democrats to hold a full vote that will formalise the impeachment inquiry in the House of Representatives. Now they got one.
It won't alter the dynamic in Washington, however.
Republicans will continue to object to what they see as an unfair process with a preordained result. Democrats will push ahead with an investigation that they always intended would culminate in dramatic public hearings and (perhaps) an impeachment vote.
This doesn't mean that Thursday's proceedings are meaningless, however. For the first time since 1998, the House is taking a significant step toward impeaching a president.
The resolution gives the public some idea what to expect in the days ahead - including lots of Intelligence Committee sparring between Democrats and Republicans, a look at the transcripts from some of the high-profile witness depositions already conducted and, at some point, a formal report that could serve as the basis of articles of impeachment.
It's not exactly uncharted territory in US history, but the course being set leads toward a historic clash with the presidency at stake.
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