Markets run by federal and state governments opened for the millions of Americans seeking insurance plans.
As many as seven million Americans who do not already have health insurance are eligible to purchase coverage in the coming months.
Republican opposition to the law led to the government shutdown on Tuesday.
At the White House on Tuesday, Mr Obama said the opportunity would be "life changing" for the "15% of Americans who don't have health insurance".
"Tens of thousands of Americans die each year just because they don't have health insurance," Mr Obama said, with Americans set to sign up for plans standing behind him.
"Millions more live with the fear that they'll go broke if they get sick. And today, we begin to free millions of our fellow Americans from that fear."
While Mr Obama and his supporters in the Democratic Party portray the law as a historic effort to extend healthcare coverage to the millions of Americans who lack it, the law's opponents say it amounts to an unprecedented intrusion into Americans' private lives.
The online health insurance marketplaces, or exchanges, struggled to meet higher than expected demand on Tuesday, with many users reporting long waiting times.
Health and Human Services spokeswoman Joanne Peters told the Associated Press news agency the agency was "prepared to make adjustments as needed [to] improve the consumer experience".
Users also reported errors while trying to create accounts and site crashes on the websites.
"The first day of something that you know is going to have a lot of bugs, it's not that frustrating," Mike Weaver, a self-employed photographer without insurance told the Associated Press.
"If it was the last day to sign up... then I'd be terribly frustrated."
Mr Obama acknowledged the glitches and delays.
"We found out that there have been times this morning where the site has been running more slowly than it normally will," Mr Obama said.
"The reason is because more than one million people visited healthcare.gov before 7:00 in the morning."
The exchanges will continue to enrol people through the end of the year for healthcare plans slated to begin next year.
The opening of the exchanges was the culmination of more than three years of political combat in Washington over the Patient Protection and Affordable Care Act, signed into law by Mr Obama in 2010 and known to both sides as Obamacare.
Republicans in Washington and in state capitals across the country have fought hard to undermine or undo the law since its passage, with the US House of Representatives taking dozens of symbolic votes to repeal it or eliminate its funding.
A legal challenge led by the law's Republican opponents ended in June 2012 when the Supreme Court validated the law's keystone provision - a requirement that Americans not receiving health coverage from their employers or the government purchase individual plans or pay a fine.
That requirement, known as the individual mandate, takes effect in January.
The healthcare law was also a central issue in the 2012 presidential election. Republican candidate Mitt Romney vowed to press for its repeal. Mr Obama handily won re-election.
On Tuesday, the US government shut non-essential services, keeping more than 700,000 workers at home, when the Democrats who control the Senate refused to agree to Republican demands to repeal the health law or delay the individual mandate.
Both Mr Obama and congressional Democrats have said they will not negotiate on Obamacare, with Mr Obama on Monday evening accusing the Republicans of attempting to refight the last election.
In spite of the political turmoil in Washington, on Tuesday Americans were able to enrol in insurance plans offered by private companies and ranging in price and level of coverage through exchanges in every state.
Sixteen states and Washington DC have opted to operate their own exchanges, while the exchanges in 34 states will be fully or partially run by the federal government.
Exchanges will also inform customers of tax subsidies available to ease the cost of the insurance. An estimated six million US citizens are expected to qualify.
Customers enrolling under Obamacare will not be covered by the insurance until 1 January however.
That corresponds with the start of the individual mandate, as well as consumer protections including a rule barring insurers from denying coverage to people with pre-existing health conditions.
Several Obamacare provisions had taken effect prior to Tuesday, including prescription drug discounts for pensioners, a rule allowing children to remain on their parents' insurance plans up to age 26, a rule barring insurers from denying coverage to children with pre-existing health conditions, and a ban on lifetime limits on health coverage.
Another major provision known as the employer mandate - which requires employers with at least 50 full-time workers to provide insurance or incur a $2,000 (Â£1,320) per employee penalty - has been delayed until 2015.
That delay, announced by the Obama administration in July, has been seen as a concession to retailers and other businesses - and an acknowledgement that the health law was not fully ready for implementation.
A central provision of President Barack Obama's healthcare reform law has taken effect, having survived Republicans' years-long effort to undermine it.