The shares shot up by more than 90 % in early trade to as high as just over $50. In the end, the stock closed with a one-day gain of 73 % at $44.90, from the initial public offering (IPO) price of $26.
But analysts cautioned that the success of the IPO was just a first step for Twitter, which must now justify its hefty valuation.
"Kudos to Twitter for orchestrating a highly successful IPO," said Lou Kerner of the Social Internet Fund.
"However, as Facebook showed, an IPO success, or disaster in Facebook's case, is really just noise in the long term. Twitter's success as a stock is going to be based on how the company performs."
Appropriately, #TwitterIPO was among the top trending topics on the social network for much of the day.
The key Twitter founders attended the opening on the New York Stock Exchange, along with "Star Trek" actor Patrick Stewart and a nine-year-old girl who operates a lemonade stand.
"Honored to join @ev @jack @biz @dickc & the @Twitter team at their historic IPO this morning," Stewart tweeted, referring to founders Evan Williams, Jack Dorsey, Biz Stone, and Twitter chief executive Dick Costolo.
"We have a lot of work ahead of us," Costolo told CNBC television from the floor of the stock exchange. "All the capital raised by this is going into the company."
Asked about Twitter's growth potential, Costolo said "it's all about making it very simple and easy for new users to come to the platform... we all have examples of why this service can be useful to everyone on the planet."
Cantor Fitzgerald analyst Youssef Squali was upbeat about the company, saying in a note to clients that "Twitter is based on a one-to-all, all-the-time broadcast distribution model, and as such, fulfills an unmet need."
"This model is highly complementary to traditional media outlets (especially TV), and fulfills the need for up-to-the-minute, trending information in real time," it said.
But Brian Wieser at Pivotal Research issued a "sell" recommendation after the opening, saying Twitter "is simply too expensive" after the hefty opening gains with "nearly the same valuation as CBS... or even Yahoo."
Larry Chiagouris, a professor of marketing at Pace University, said the "investor mania" around Twitter is not an indication of success.
"The fundamental question is how much people have to say on Twitter," he told AFP.
"We know there are some people who are social and want to talk all the time, but you can't make a business model on those people."
Chiagouris added that "large corporations with hundreds of millions of dollars have not put substantial sums into paid media with Facebook and Twitter."
"They all are experimenting but nobody is putting 25 % in social media. It may not sound cool but traditional media is still the media of choice today," he said.
Twitter offered 70 million shares trading under the symbol TWTR, generating $1.82 billion, and gave underwriters a 30-day option to purchase an additional 10.5 million shares.
The IPO assigned a market value of around $14.4 billion to the company whose messaging service has become a hugely popular tool for celebrities, journalists, political leaders and others. But by the end of the day, that value had topped $24 billion.
With the over-allotment it should be the second-biggest tech IPO after Facebook's $16 billion effort last year and ahead of Google's 2004 offer, which raised $1.92 billion, according to research firm Dealogic.
Depending on the outcome of the common stock offer to underwriters, between 12.8 and 14.5 % of the company's shares will be publicly traded. The rest are held by its founders and a handful of early investors.
Twitter has fast become engrained in popular culture but must still convince investors of its business model, having lost more than $440 million since 2010.
But with 232 million users and growing, Twitter is expected to be able to reach profitability by delivering ads in the form of promoted tweets, and from its data analytics.
The research firm eMarketer estimates Twitter will bring in $582.8 million in global ad revenue this year, and nearly $1 billion in 2014.