Updated

The Obama administration said Sunday a late surge in sign-ups resulted in 1.1 million Americans enrolling in ObamaCare before last week’s deadline to get insurance coverage starting Jan. 1.  -- a sign the federal website is getting fixed but not enough good news to sway critics of the health care law.

“A million Americans?” Rep. Darrell Issa, R-Calif., said to Fox News. “I don’t think that’s anything to celebrate. It was a failed [website] launch, a flawed law and it needs real change.”

Rep. Joaquin Castro, D-Texas, acknowledged problems but supported the legislation and argued every big law has needed early changes.

“It gets health care to people in a better way,” he said on NBC’s “Meet the Press.”

The administration said more than 975,000 Americans signed up between December 1 and the December 24 deadline to ensure coverage beginning on the first day of 2014.

Officials also boasted HealthCare.gov was able to support 83,000 concurrent visitors on December 23, the next-to-last day before the sign-up deadline.

The numbers released Sunday are indeed a dramatic improvement from the mere 137,000 Americans who enrolled in November and the 27,000 who signed up in October, the first month of operation for the error-prone site.

However, Dr. Scott Gottlie, of the American Enterprise Institute, questions the extent of the late surge, considering 900,000 had enrolled by Dec. 22. And he told “Fox News Sunday” that he expected the total number to be roughly 2 million when he came on the show.

The enrollment numbers come at a critical moment for President Obama's sweeping health care law, which becomes "real" for many Americans on Jan. 1 when coverage through the exchanges and key patient protections kick in.

"As we continue our open enrollment campaign, we experienced a welcome surge in enrollment as millions of Americans seek access to affordable health care coverage," Marilyn Tavenner, the head of the Center for Medicare and Medicaid Services, said in a blog post.

The administration has yet to provide a December update on the 14 states running their own exchanges. While California, New York, Washington, Kentucky and Connecticut have performed well, others are still struggling.

The fledgling federal and state exchanges are still likely to fall short of the government's own targets for 2013.  The administration had projected more than 3.3 million overall would be enrolled through both by the end of the year.

Tavenner said fixes to the website, which underwent a major overhaul to address widespread outages and glitches, contributed to December's figures. But the problems haven't totally disappeared. Thousands of people wound up waiting on hold for telephone help on Christmas Eve for a multitude of reasons, including technical difficulties.

For Americans who successfully chose insurance plans by Dec. 24, coverage should start on New Year's Day for those who pay their first month's premium by the due date, which in most cases has been extended until Jan. 10.

But insurers have complained that another set of technical problems, largely hidden from consumers, has resulted in the government passing along inaccurate data on enrollees. The White House says the error rate has been significantly reduced. Yet with a flood of sign-ups that must be processed in just days, it remains unclear whether last-minute enrollees will encounter a seamless experience if they try to use their new benefits come Jan. 1.

The political fallout from the website's calamitous rollout could pale in comparison to the heat that Obama might take if Americans who signed up and paid their premiums arrive at the pharmacy or the emergency room and find there's no record of their coverage.

Republican critics, already on the lookout for health-law failures to exploit in the 2014 midterm elections, would be emboldened to argue that shortcomings with the law's implementation have jeopardized Americans' health.

As make-or-break January approaches, officials are also working to prevent gaps in coverage for millions of Americans whose individual policies were canceled this fall because they fell short of the law's requirements. In one of a series of last-minute tweaks, the administration in December said even if those individuals don't sign up for new plans, they won't face the penalty the law imposes on Americans who fail to get insurance by March 31.

A key indicator of whether state-run exchanges are keeping pace with the federal exchange will come next month, when the administration releases full December figures. Overall, the goal is to sign up 7 million Americans before the first-year open enrollment period closes at the end of March.

A few states offering their own updates have posted encouraging totals, including New York, where more than 200,000 have enrolled either through the state exchange or through Medicaid, a government program expanded under Obama's health law to cover more people. In California, a tally released Friday showed nearly 430,000 have enrolled through the exchange so far.

Another major unknown is whether the recent surge in enrollments skewed toward older Americans whose medical needs are expensive to cover, or whether the administration succeeded in recruiting younger and healthier people whose participation is critical to the law's success. Those details for December are expected to be released in mid-January.

Former Vermont Gov. Howard Dean, a Democrat, agreed with Gottlie on “Fox News Sunday” that the high number of expensive policies purchased suggests older or less-healthy people are enrolling because such plans provide more extensive coverage.

The Associated Press contributed to this report.