Japan's Nikkei average hit a one-week low before erasing losses to stand almost flat by midday.
MSCI's broadest index of Asia-Pacific shares outside Japan managed to tick up 0.2% as firmness in Chinese shares countered global weakness.
Still, investors were cautious after the US S&P 500 lost 2.7% last week, its biggest weekly decline since the week ending June 1, 2012, to hit two-month lows, with financials leading the decline.
Leading losses in the past week were European shares, with German shares falling 4.5% and French shares 3.3%, hurt by concerns over tension between Russia and the West as well as losses at Banco Espirito Santo , the biggest bank in Portugal.
Portugal said on Sunday it will spend 4.9 billion euros ($6.58 billion) to rescue Banco Espirito Santo, testing the euro zone's resilience to another banking crisis just months after Lisbon exited an international bailout.
Investors also have been worrying about the impact of sanctions against Russia. About 40 European blue-chips, including many German companies, derive more than 5% of their revenues from the Russian market.
"The impact of sanctions can be big when consumption is not so strong worldwide. US consumption is perhaps okay, but Japan is weak and in Europe even Germany seems to be losing steam. I think the market is still under-evaluating the impact of the sanctions on Russia," said a prop trader at a Japanese bank.
Argentine's debt default last week also added to the gloom, even though it has so far had limited spillover to any other emerging markets.
"The fact that US financial shares fell sharply even though they have very limited exposure to Argentine and Portugal suggests just how markets are getting nervous about high valuations," said Yasuo Sakuma, portfolio manager at Bayview Asset Management.
END OF GOLDILOCKS?
World share markets had rallied for much of this year on hopes that the US economic growth will pick up while at the same time the Fed will also maintain zero interest rates at least until the middle of next year to support the economy.
Friday's US job data provided little reason for the Fed to hurry in raising rates.
Although the closely-watched monthly payroll gains topped the 200,000 mark for six months in a row, the unemployment rate rose to 6.2% and average hourly earnings rose only one cent, showing little inflationary pressure.
Still, some analysts say the data portends possible risk for US shares as it shows US growth is growing but hardly accelerating.
"One reason there's no sign of wage inflation is because job gains in the past year has been concentrated on low-paid jobs," said Tohru Yamamoto, chief fixed income strategist at Daiwa Securities.
"So overall, yes, the data was solid. But it shows no sign of acceleration, which will spell troubles for stocks as their valuation is based on the assumption that growth will pick up," he added.
US Treasury debt prices rose as traders trimmed bets the Fed would push rates up in the first half of next year.
The rate-sensitive two-year notes yield fell more than five basis points to around 0.480%. The 10-year yield also dropped to 2.505%, off three-week high of 2.614%, hit on Thursday.
As US debt yields fell back, the dollar took a breather, with its index against a basket of major currencies off a 10-1/2-month high hit on Thursday.
The dollar index stood at 81.343, down from Thursday's high of 81.573.
"The July jobs data won't change the Fed's benign stance as it was about as 'goldilocks' as it could be," said Shane Oliver, Head of Investment Strategy at AMP Capital in Sydney.
The euro fetched $1.3422, little changed in early trade but off last week's 8-month low of $1.3366. The dollar stood at 102.64 yen, down from 103.15 yen, a four-month peak hit on Wednesday.
Oil prices were under pressure as oversupply in the Atlantic basin and low demand outweighed worries over political tensions in the Middle East, North Africa and Ukraine.
US crude futures traded at $97.93 per barrel, after having hit a six-month low of $97.09 on Friday.
The market has so far shown muted response to the news that Islamic State fighters seized control of Iraq's biggest dam, an oilfield and three more towns on Sunday after inflicting their first major defeat on Kurdish forces since sweeping across much of northern Iraq in June.