And it would end the tax penalty Obama's statute imposes on people who don't buy insurance. President Donald Trump recently called the measure "mean" and has since been asking for a more compassionate Senate bill.
Democrats gathered on the Senate floor and defended Mr Obama's 2010 overhaul.
Four GOP conservative senators - Ted Cruz of Texas, Mike Lee of Utah, Kentucky's Rand Paul and Ron Johnson of Wisconsin - said the bill falls short of GOP promises to erase Mr Obama's law and lower people's costs.
Indiana Republican Todd Young says he is still reading the bill, but says in a statement that "doing nothing is not and option" and that "Obamacare has failed Hoosiers as prices have skyrocketed".
Trump says at the start of a White House event on technology he is hopeful Congress will get something done on health care "with heart". This bill will also significantly impede our efforts to combat the opioid and behavioral health crises Virginia and many other states face by jeopardizing coverage for people who are struggling with addiction. The additional funds would continue through 2020, and be gradually reduced until they are entirely eliminated in 2024. He said "small tweaks" during the upcoming debate "cannot change the fundamental meanness at the core of this legislation".
-Congressional Budget Office analysts review the bill and its effect, both on Americans and the nation's finances. Some from states that have expanded the program have battled to prolong the phase-out, while conservative Republicans have sought to halt the funds quickly.
Under Obamacare, the amount of assistance available was linked to how much people earn and where they live, with larger subsidies for lower-income consumers and residents of regions with very high insurance costs.
The bill, with the Orwellian-sounding name "Bettercare", can be viewed in its entirety at the Senate's Committee on Budget website.
U.S. Sen. Chris Murphy, D-Conn., said, "No surprise that insurance lobbyists are helping write this bill". States would also have to retain Obama's requirement that family insurance cover children up to age 26.
Like the House bill, the Senate bill would defund Planned Parenthood for one year. But Republicans won over skeptical members with last-second amendments and passed the bill by a narrow margin. In addition, it restricts federal subsidies for being used on "any health plan that includes coverage for abortions" with, again, similar exceptions.