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For all those who have doubted its business acumen, Netflix had a resounding answer on Wednesday: 27.15 million.

That’s the number of American homes that were subscribers to the streaming service by the end of 2012, beating the company’s own projections for the fourth quarter after a couple of quarters of underwhelming results.

Netflix’s growth spurt in streaming — up by 2.05 million customers in the United States, from 25.1 million in the third quarter — was its biggest in nearly three years, and helped the company report net income of $7.9 million, surprising many analysts who had predicted a loss.

The results reflected just how far Netflix has come since the turbulence of mid-2011, when its botched execution of a new pricing plan for its services — streaming and DVDs by mail — resulted in an online flogging by angry customers. Investors battered its stock price, sending it from a high of around $300 in 2011 to as low as $53 last year.

“It’s risen from the ashes,” said Barton Crockett, a senior analyst at Lazard Capital Markets. “A lot of investors have been very skeptical that Netflix will work. With this earnings report, they’re making a strong argument that the business is real, that it will work.”

Investors, cheered by the results, sent Netflix shares soaring. By Thursday afternoon the shares were up more than 37 percent to $141.49.

Netflix’s fourth-quarter success was a convenient reminder to the entertainment and technology industries that consumers increasingly want on-demand access to television shows and movies. Streaming services by Amazon, Hulu and Redbox are all competing on the same playing field, but for now Netflix remains the biggest such service, and thus a pioneer for all the others.

“Our growth and our competitors’ growth shows just how large the opportunity is for Internet TV, where people get to control their viewing experience,” Netflix’s chief executive, Reed Hastings, said in a telephone interview Wednesday evening.

Questions persist, though, about whether Netflix will be able to attract enough subscribers to keep paying its ever-rising bills to content providers, which total billions of dollars in the years to come. The company said on Wednesday that it might take on more debt to finance more original programs, the first of which, the political thriller “House of Cards,” will have its premiere on the service on Feb. 1. Netflix committed about $100 million to make two seasons of “House of Cards,” one of five original programs scheduled to come out on the service this year.

“The virtuous cycle for us is to gain more subscribers, get more content, gain more subscribers, get more content,” Mr. Hastings said in an earnings conference call.

The company’s $7.9 million profit for the quarter represented 13 cents a share, surprising analysts who had expected a loss of 12 cents a share. The company said revenue of $945 million, up from $875 million in the quarter in 2011, was driven in part by holiday sales of new tablets and television sets.

Netflix added nearly two million new subscribers in other countries, though it continued to lose money overseas, as expected, and said it would slow its international expansion plans in the first part of this year.

The “flix” in Netflix, its largely forgotten DVD-by-mail business, fared a bit better than the company had projected, posting a loss of just 380,000 subscribers in the quarter, to 8.22 million. The losses have slowed for four consecutive quarters, indicating that the homes that still want DVDs really want DVDs.

On the streaming side, Netflix’s retention rate improved in the fourth quarter, suggesting growing customer satisfaction.

Asked whether the company’s reputation had fully recovered after its missteps in 2011, Mr. Hastings said, “We’re on probation at this point, but we’re not out of jail.”

He has emphasized subscriber happiness, even going so far as to say on Wednesday that “we really want to make it easy to quit” Netflix. If the exit door is well marked, he asserted, subscribers will be more likely to come back.

The hope is that original programs like “House of Cards” and “Arrested Development” will lure both old and new subscribers to the service. Those programs, plus the film output deal with the Walt Disney Company announced in December, affirm that Netflix cares more and more about being a gallery — with showy pieces that cannot be seen anywhere else — and less about being a library of every film and TV show ever made.

“They’re morphing into something that people understand,” said Mr. Crockett of Lazard Capital.

Mr. Hastings said this had been happening for years, but that it was becoming more apparent now to consumers and investors.

Mr. Hastings’s letter to investors brought up the elephant in the room, the activist investor Carl C. Icahn, who acquired nearly 10 percent of the company’s stock last October. Mr. Icahn, known for his campaigns for corporate sales and revampings, stated then that Netflix “may hold significant strategic value for a variety of significantly larger companies.”

Netflix subsequently put into place a shareholder rights plan, known as a poison pill, to protect itself against a forced sale by Mr. Icahn.

The company said on Wednesday, “We have no further news about his intentions, but have had constructive conversations with him about building a more valuable company.”

Factoring in the stock’s 30 percent rise since November and the after-hours action on Wednesday, Mr. Icahn’s stake has now more than doubled in value, to more than $700 million from roughly $320 million.