* Stoltenberg to take over on Oct. 1
* NATO faces new challenge from Russia
* After Afghanistan, NATO to refocus on Europe (Updates with Stoltenberg quotes, reaction)
By Adrian Croft and Balazs Koranyi
BRUSSELS/OSLO, March 28 (Reuters) - NATO chose former Norwegian Prime Minister Jens Stoltenberg as its next leader on Friday at a time when the Western military alliance must deal with a resurgent Russia following its annexation of Ukraine's Crimea.
Stoltenberg will take over as secretary-general of the 28-nation grouping on Oct. 1, succeeding former Danish prime minister Anders Fogh Rasmussen, who has led NATO since 2009.
Stoltenberg, the first Norwegian to occupy NATO's top post, will take over at a time when NATO, seen by some as a Cold War relic, has gained new relevance because of concerns about what the Ukraine crisis says about a newly assertive Russia.
Stoltenberg said the Ukraine crisis "reminds us just how important NATO is. The idea of collective defence has become more important given how Russia is using force to change borders in Europe."
"Russia must see that what they've done carries a price," he said at a news conference in Oslo.
However, he said the Ukraine crisis would not turn the clock back to the Cold War.
"The current situation is not that bad," he said.
Daniel Keohane, a defence expert at the FRIDE thinktank, said Norway was seen as a very serious defence player. "(It) has always taken the challenge of Russia very, very seriously. I think there is a little bit of a signal there," he said.
Stoltenberg, 55, who was backed by the United States, Germany and Britain, will take over at a turning point in NATO's history.
The urgency of the Ukraine crisis means that the alliance, which is due to end combat operations in Afghanistan at the end of this year, is likely to refocus back onto its core task of defending its member countries.
Unusually, NATO has appointed a leader from a country outside the European Union. But diplomats said that should not hinder Stoltenberg fostering cooperation between NATO and the EU, which has taken on a growing security role.
(For a newsmaker on Stoltenberg, click on )
By appointing Stoltenberg early, NATO has ensured that the contest for the top alliance post is not tangled up with horse-trading for top jobs in the EU, including European Commission president and EU foreign policy chief, which fall vacant later this year. The United States had been keen to avoid that happening, analysts said.
Traditionally, NATO's top political job is held by a European while the top military post goes to an American.
The United States welcomed the decision.
"Mr. Stoltenberg is a proven leader with a demonstrated commitment to the transatlantic alliance ...We are confident he is the best person to ensure the continued strength and unity of the NATO alliance," a White House statement said.
British Prime Minister David Cameron said he was delighted by the choice.
"Recent events in Ukraine have underlined that, even once we complete our mission in Afghanistan, there will be new challenges to respond to," he said in a statement.
German Foreign Minister Frank-Walter Steinmeier said Stoltenberg was well suited to position NATO strategically "in the face of big new tasks such as in crisis management or cyber security."
NATO foreign ministers are expected to discuss next week how to reinforce the alliance's military presence in nervous eastern European countries such as the Baltics and Poland.
Stoltenberg also faces a challenge in trying to persuade European countries to reverse, or at least end, sharp cuts in defence spending that many of them have pushed through in response to the financial crisis.
U.S. President Barack Obama voiced concern during a visit to Europe this week over defence cuts by some NATO allies and said the alliance would have to look at whether "everybody is chipping in."
Stoltenberg, who served for nearly 10 years as Norway's prime minister before losing elections last September, reinforced that message, saying: "We have to understand that military spending is important even in times of austerity and the Russian military buildup in Ukraine reminds us of that."
However, in a sign of obstacles he will face in convincing allies to spend more on defence, Dutch Finance Minister Jeroen Dijsselbloem poured cold water on the idea on Friday, telling reporters in The Hague: "More money means we would need to make more spending cuts (elsewhere). I can't imagine there's much enthusiasm for that. It would not be sensible."
Others in the frame to succeed Rasmussen included and former Italian foreign minister Franco Frattini, whose bid had been supported by the Italian government.
Italian Foreign Minister Federica Mogherini told reporters in The Hague earlier this week that the next NATO leader would have to take a personal interest in the Mediterranean region.
Obama discussed the top NATO job with the Italian government during a visit to Rome this week, succeeding in winning over Rome to accepting Stoltenberg.
Stoltenberg will take over the helm soon after NATO's Sept. 4-5 summit in Wales which will mark the imminent ending of NATO-led combat operations in Afghanistan.
He is considered a skilful economic operator who got Norway through the global financial crisis relatively unharmed as the government used its stored oil wealth to boost spending, create demand and keep unemployment low.
His governments backed NATO's military campaigns in Afghanistan and Libya.
Stoltenberg was Norway's prime minister when far right militant Anders Behring Breivik went on a bombing and shooting spree in 2011 that killed 77 people. (Additional reporting by James Mackenzie in Rome, Michelle Martin in Berlin, Guy Faulconbridge and William James in London, Anthony Deutsch in Amsterdam, Jeff Mason in Washington, Editing by Angus MacSwan)