The nearly party-line 51-48 vote came on a nonbinding Republican-backed budget measure that eases the way for action on subsequent repeal legislation as soon as next month.
"We must act quickly to bring relief to the American people," said Senate Majority Leader Mitch McConnell, R-Ky.
The House is slated to vote on the measure on Friday, though some Republicans there have misgivings about setting the repeal effort in motion without a better idea of the replacement plan.
"We're going to do repeal and replace, very complicated stuff," Trump told reporters, adding that both elements would pass virtually at the same time. That promise, however, will be almost impossible to achieve in the complicated web of Congress, where GOP leaders must navigate complex Senate rules, united Democratic opposition and substantive policy disagreements among Republicans.
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Passage of today's measure would permit follow-up legislation to escape the threat of a filibuster by Senate Democrats. Republicans are not close to agreement among themselves on what any "Obamacare" replacement would look like, however.
The 2010 law extended health insurance to some 20 million Americans, prevented insurers from denying coverage for pre-existing conditions and steered billions of dollars to states for the Medicaid health program for the poor.
Republicans fought the effort tooth and nail and voter opposition to Obamacare helped carry the party to impressive wins in 2010, 2014, and last year.