Roberts will preside over the trial, which is scheduled to formally begin Tuesday.
The swearing in came after two articles of impeachment were introduced in the Senate, marking the start of a historic proceeding that could potentially remove Trump from office.
The seven Democratic managers tasked with presenting the evidence at trial read the articles on the Senate floor. They accuse Trump of abusing his power by trying to press Ukraine to investigate Democratic presidential candidate Joe Biden, and obstructing the House investigation that followed.
Although they mark the official opening of the trial, Thursday's activities are entirely procedural. The meat of the trial won't begin until Tuesday.
The Senate is expected to formally notify the White House and "summon the president to answer the articles and send his counsel."
After the preliminary steps are completed, the Senate will table an initial resolution outlining how it will move forward with the trial. McConnell has said that measure will establish periods for arguments by House managers and Trump's legal team, and as well as set times for senators to submit written questions.
Still undecided is whether witnesses will be allowed to testify at trial. McConnell and his Republican majority caucus are opposed. Pelosi and Democratic leader Chuck Schumer have lobbied for weeks that the Senate trial include witnesses and new evidence that's emerged since the House passed the articles Dec. 18.
Democrats need four Republicans to cross the aisle in a vote to include witnesses. Sen. Mitt Romney, R-Utah, has said he wants to hear from former national security adviser John Bolton and would vote for him to testify, while Sens. Susan Collins of Maine and Lisa Murkowski of Alaska, both Republicans, are also viewed as potential swing votes.
A conviction on either charge, which is considered unlikely by most experts, requires a two-thirds majority vote in the Senate.
Earlier Thursday, the Government Accountability Office said in a report the Trump administration broke U.S. law last year when it withheld Congress-approved military aid to Ukraine -- the issue that ultimately spawned the impeachment investigation.