* E.ON sells its 3.5 pct Gazprom stake
* E.ON set to book gain of 2.5 bln eur
* E.ON shares extend gains to 1.3 percent
* Gazprom shares rise to highest since April
* E.ON sold shares at below market price - analyst
(Adds details on E.ON's strategy, graphic)
By Peter Dinkloh and John Bowker
FRANKFURT/MOSCOW, Dec 1 (Reuters) - Germany's E.ON (EONGn.DE), the world's largest utility, agreed to sell its 3.5 percent stake in Russian gas monopoly Gazprom (GAZP.MM) for 3.4 billion euros ($4.5 billion) in a first step to raise money to expand outside Europe and to cut debt.
The Duesseldorf-based company sold 2.7 percent to Russian state bank VEB and 0.8 percent on the stock market, E.ON said on Wednesday. E.ON shares extended gains and Gazprom stock rose to its highest since April.
For graphics on global mergers & acquisitions click on:
"VEB is a financial investor, so maybe there will be discussions coming up about higher dividends," said Ildar Davletshin, an analyst at Renaissance Capital in Moscow.
"There has been talk that Gazprom could be broken up, but they would not buy a stake in a company they do not see a future in. The government isn't planning anything bad."
According to Davletshin's calculations, E.ON had sold the stake at below market value -- approximately 165 roubles a share, compared with Tuesday's market close of 174.2 roubles ($5.53).
E.ON is seeking to lower its 45 billion euro debt burden and raise money to expand outside the continent as earnings in Europe drop due to a tax on its nuclear power plants in Germany and demand falls because of the economic slowdown.
The company plans to generate 15 billion euros through divestments in the next three years and aims to identify two regions for investing part of the money.
E.ON Chief Executive Johannes Teyssen has said the company is still considering which regions it wants to expand into but declined to give details.
German magazine Der Spiegel reported he was examining China and Brazil.[ID:nLDE6A507S]
E.ON shares extended gains and were up 1.3 percent to 22.39 euros at 1554 GMT, compared with a 0.4 percent increase in the Stoxx Europe 600 utilities index .SX6P. Gazprom stock rose 4.8 percent to 182.49 roubles.
Writing by Peter Dinkloh; Editing by Jonathan Gould, Matthew Jones and David Hulmes