Up to 800,000 public service workers were placed on temporary unpaid leave and all but non-essential government activities were suspended - for the first time since the winter of 1995/96.
Tourists were also hit after a midnight deadline on the budget expired as the action caused the closure of national parks and museums along the Washington Mall and US Capitol visitor centres.
Mr Obama urged lawmakers to vote to keep government operations running and to raise the nation's borrowing cap without conditions. He said: "They've shut down the government over an ideological crusade to deny affordable health insurance to millions of Americans.
"Many Representatives have made it clear that had they been allowed by Speaker [John] Boehner to take a simple up or down vote on keeping government open with no strings attached, enough votes from both parties would have kept the American people's government open and operating."
The President also warned Republicans against using a crucial mid-October deadline to raise the government's $16.7 trillion debt ceiling as leverage to try to reverse the healthcare law or achieve other political objectives.
"Congress, generally, has to stop governing by crisis," he said. "I'm not going to allow anybody to drag the good name of the USA just to re-fight a settled election or extract ideological demands."
A debt default that would result if Congress fails to raise the debt ceiling when it is reached in less than three weeks could be devastating, Mr Obama said. The threat of default in 2011 resulted in a painful debt rating downgrade, he added. "If they go through with it this time, and force the US to default for the first time in its history, it would be far more dangerous than a government shutdown, as bad as a shutdown is. It would be an economic shutdown."
Federal employees, whose work has been labelled non-essential, have been hit hard as political dysfunction repeatedly stifles negotiations between Democrats, who control the Senate, and Republicans, who lead the House of Representatives.
Mr Obama wrote to the workers, saying they do valued work "in a political climate that, too often in recent years, has treated you like a punching bag".
Whether furloughed employees would get paid remained unclear. House members from Maryland and Virginia said they had introduced a bill to require all federal employees to receive retroactive pay for the shutdown, which is what happened in 1996.
If employees do not get paid, at least one administration official said he would reduce his own salary. US Attorney General Eric Holder said he would take a pay cut equivalent to the largest any employee at the Justice Department faced.
Similarly, Republican Senator Ted Cruz, who helped spur the shutdown by urging his party leaders to demand delays to the rollout of Obama's healthcare plan, said he would donate his salary to charity for each day of the shutdown.