At 0955 GMT, the FTSEurofirst 300 index of leading European shares was up 2.1 percent at 833.00 points, and had been as high as 839.86. The index has lost 21.6 percent in October, hurt by a credit crisis and recession worries.
Volkswagen was up 64 percent, following its 146 percent surge on Monday. Short sellers piled into the stock to sew up their speculative positions after Porsche bought up nearly all VW's remaining free float.
BP rose 4.2 percent after it reported a 148 percent rise in third-quarter replacement cost profit, at $10.03 billion, boosted by higher oil prices.
Total, ENI, and Royal Dutch Shell were up between 0.3 and 3.1 percent. "It's no real surprise that bargain-hunters are coming into these markets, despite the fact that expectations for the economy are tumbling and the outlook on the corporate front is gloomy," said Henk Potts, strategist at Barclays stockbrokers.
"There's too much bad news priced into these markets. Long-term investors can look through the dark clouds ... and can see the cheap valuations."
BG Group was up 2 percent after launching a A$5.6 billion ($3.4 billion) friendly takeover bid for Australia's Queensland Gas Co Ltd(QGC) as it tries to secure gas to boost its position in Asia's lucrative liquefied natural gas market.
Most miners were higher as metals prices in London and Shanghai recovered after early falls, bouncing off lows on the back of losses by the dollar ahead of a U.S. central bank meeting later in the day.
London Metal Exchange copper for delivery in three months rose 0.75 percent to $4,030, having dipped almost 5 percent earlier.
BHP Billiton, Rio Tinto, and Xstrata were up between 3.3 and 6.8 percent.
But Eurasian Natural Resources Corp. and Kazakhmys were down 5.9 and 4 percent respectively after the Kazakh government said the two companies would reduce output due to the global economic turmoil.
Later in the session, the U.S. Federal Reserve starts a two-day meeting expected to cut the fed funds rate -- now at 1.50 percent -- by at least a further 50 basis points.
"That should set the tone for the Bank of England and the ECB to cut rates," said Potts of Barclays. "The inflation picture has improved."
Britain's FTSE 100 was up 2.1 percent, Germany's DAX was up 9 percent, boosted by Volkwagen, and France's CAC-40 was up 0.3 percent.
Aviva surged 10.8 percent after the insurer said it had had no discussions with the UK government about capital support and reported a 12 percent rise in sales for the nine months to September.
Its rival Aegon was up 0.9 percent after saying that the Dutch government would provide 3 billion euros capital. The company said it will scrap its final 2008 dividend, as it reported a third-quarter loss of 350 million euros.
Terms of the injection are nearly identical to the deal between the Dutch government and financial group ING announced last week.
ING was one of the biggest losers in the index on Tuesday, down 8.9 percent, after Fitch cut its outlook to "negative" from "stable".
Other banks that fell included Deutsche Postbank, down 9.7 percent after JP Morgan slashed its price target to 16.5 euros from 52.3 euros while maintaining a "neutral" rating.
French bank Societe Generale was down 2.2 percent despite moving to reassure investors after a sharp fall in its share price on Monday, saying it was sticking by its profit forecast and had no bad surprises in its market operations.
Spain's biggest bank, Santander, was up 2 percent after posting a 5.5 percent rise in net profit for the first nine months to September, underpinned by robust recurrent earnings from its core retail banking business.