In Europe, Britain's FTSE 100 rose 0.9 percent to close at 6,802.92 while France's CAC 40 rose by the same rate to 4,461.12. Germany's DAX advanced 0.7 percent to 9,902.41.
Shares in BNP Paribas, France's biggest bank, rose 3.6 percent after it said would be able to handle the U.S. fine it got for violating sanctions. The bank processed billions of dollars in illegal transactions on behalf of clients in Sudan, Cuba and Iran that the U.S. had blacklisted to block their participation in the global financial system. The case had weighed on BNP's share price and the resolution, though it involves a massive fine, is a relief for shareholders.
In the U.S., the Dow was up 0.8 percent at 16,958.27 and the S&P 500 was up 0.7 percent at 1,973.72.
Earlier, in Asia, trading was subdued as investors appeared to shrug off the Chinese manufacturing data.
Mainland China's key benchmark, the Shanghai Composite Index, added 0.1 percent to 2,050.38. Markets in Hong Kong were closed for a public holiday.
Japan's Nikkei 225 gained 1.1 percent to 15,326.20 after the central bank released a survey showing better-than-expected business sentiment despite a decline in the April-June quarter.
Elsewhere, shares in South Korea and Australia also fell, while those in Taiwan and New Zealand gained.
In energy trading, benchmark U.S. crude for August delivery was down 29 cents to $105.09 per barrel in electronic trading on the New York Mercantile Exchange. The contract fell 37 cents on Monday.
In currencies, the dollar rose to 101.50 yen from 101.32 yen late Monday. The euro slipped to $1.3680 from $1.3693.