A brighter mood sets in at the New York Stock Exchange
Relief that the fatal crash of an American Airlines plane in New York the previous day looked less likely to have been caused by terrorism also fuelled the rises.
Wall Street's leading indexes rallied from the outset, giving an extra boost to gains in London, Frankfurt and Paris.
In New York, the Nasdaq and Dow Jones built on their early gains to close almost 3% higher at 1892 in the case of the tech-heavy Nasdaq, and up 2% at 9,750 for the Dow.
The FTSE 100 index of blue chip stocks closed the day 2.5% higher at 5277.
Relief
The prospect of a swift, firm victory for the US-led military campaign in Afghanistan cheered investors.
It sparked hopes that American consumers might soon regain the confidence to spend, thus easing the economic slowdown.
"Investors are certainly acting like (the war) is not going to be a protracted drag on economic activity and therefore profit growth," said Charlie Crane of Victory SBSF Capital Management.
Airline stocks rally
Shares in major US and European airlines continued to rebound after Monday's plunge, with Air France, Lufthansa, British Airways all closing firmer.
Among US peers, shares AMR, in the parent company of American Airlines which operated the crashed plane, climbed 3%.
Hotel and tour industry stocks also did well, reversing some of Monday's losses.
Tech cheer
But technology stocks put in a stellar performance on both sides of the Atlantic as investors bet on a brighter outlook for the troubled sector.
US computer firm Compaq rose 15% on a positive analyst's note, while its merger partner Hewlett-Packard climbed 5%.
Among others, US semiconductor-maker Intel gained, while German chip firm Infineon added 10%.
On the London market, strong earnings for UK telecoms giant Vodafone pulled others up with it.
Leeway
Relief over Afghanistan and the lifting of the cloud over the airline industry also gave investors a bit of leeway to examine other issues.
The reluctance to see Afghanistan's Northern Alliance push on and take Kabul unilaterally - which was recently voiced by both the United States and British governments - could not dampen the market's surge.
"Today the political concerns have abated a little bit," said Rick Meckler of investment firm LibertyView in New York.
"It has allowed those people who would prefer to focus on the economic stimulus in the market to go out and buy," he said.
The US central bank has cut interest rates 10 times this year in a bid to get the economy moving.