Markets in mainland China remained closed as the U.S. warned against all travel to China following the WHO's announcement.
Japan's Nikkei 225 index rose 1% to 23,210.44 while the S&P ASX/200 edged 0.2% higher to 7,021.00. Hong Kong's Hang Seng also rose 0.2%, to 26,507.08 and the Sensex in India climbed 0.4% to 41,068.92. Taiwan's benchmark picked up 1.1%, while South Korea's Kospi declined 0.2% to 2,144.03. Shares also fell in Jakarta and Kuala Lumpur.
A monthly survey of manufacturers in China showed factory activity weakened, but it did not reflect the latest developments from the virus outbreak.
A late wave of buying left major U.S. stock indexes with modest gains on Wall Street after they spent most of the day in the red. Technology companies and banks fared best.
China reported 9,692 confirmed cases on Friday with a death toll of 213, including 43 new fatalities.
Investors worry that the outbreak could end up dampening global economic growth. But those concerns appeared to ease by late afternoon, after the director general of the World Health Organization said that the organization was not recommending limiting travel or trade to China.
"There is no reason for measures that unnecessarily interfere with international travel and trade," Tedros Adhanom Ghebreyesus told reporters in Geneva after the WHO officially declared the outbreak a global emergency.
The S&P 500 index rose 0.3% to 3,283.66 after falling as much as 0.9% earlier in the day. The Dow Jones Industrial Average climbed 0.4% to 28,859.44, while the Nasdaq added 0.3% to 9,298.93. Smaller company stocks recovered most of the way after taking the brunt of the selling. The Russell 2000 index slipped 0.1% to 1,648.22.
Stocks have given up some ground after a strong start to the year amid uncertainty over the virus outbreak. Still, the major indexes remain on track to end January with gains.
The WHO's move to declare the outbreak a global emergency Thursday came after the number of cases spiked tenfold in a week. The declaration means the WHO sees the virus as a risk to other countries that requires an international response.
"While the WHO declaration was anticipated to come at some point, they did stop short of suggesting travel and trade restrictions with China were necessary to prevent the spread of the virus," said Mike Stritch, chief investment officer of BMO Wealth Management. "The aversion of a 'worst case scenario' put a floor under equities with airlines, for example, moving higher in the afternoon."
Companies have been issuing warnings over the potential impact to profits and revenue from the outbreak. Align Technology, which makes tooth-straightening systems, gave investors a weak profit forecast because of the virus. Starbucks has already held back on raising its forecast for the year and airlines are starting to curtail flights to Chinese cities because of weak demand.
Jitters over the virus outbreak had many investors initially seeking less risky assets Thursday. That drove up the prices of U.S. government bonds and gold. The yield on the 10-year Treasury note fell to 1.55% from 1.59% late Wednesday. It rebounded to 1.58% by Friday morning.
The price of gold fell $12.20 to $1,571.30 per ounce. Gold prices are up 20% over the past year.
Tesla surged 10.3% after the electric vehicle maker blew past Wall Street's fourth-quarter earnings forecasts on record sales. The company also told investors that it is ramping up production of the Model Y small SUV, which is a key product because consumers are buying smaller utility vehicles.
Other companies failed to impress traders, however.
Altria slid 4.2% after the maker of Marlboro cigarettes reported hefty costs because of its investment in e-cigarette maker Juul. Altria took a 35% stake in Juul at the end of 2018 and that company has since faced a surge in federal and state investigations into its marketing amid an explosion of underage vaping teenagers.
Shares in UPS skidded 6.7% after the package delivery company gave investors a disappointing profit forecast.
Benchmark crude oil rose $1.10 to $53.24 per barrel in electronic trading on the New York Mercantile Exchange. It fell $1.19 to settle at $52.14 a barrel on Thursday. Brent crude oil, the international standard, picked up $1.11 to $58.44 per barrel. It dropped $1.58 to close at $58.33 a barrel overnight.
In other commodities trading, silver rose 50 cents to $17.99 per ounce and copper fell 3 cents to $2.52 per pound.
The dollar rose to 109.08 from 108.98 yen on Thursday. The euro fell to $1.1024 from $1.1033.