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* Saudi to choose when to phase out reduction in coming months

* Brent, U.S crude futures hit highest since January 2020

* Russia allowed small increase to meet local demand

* Yemen's Houthis say they fired missile at Saudi Aramco site

(Updates with settlement prices, adds graphic and details)

By Arpan Varghese

BENGALURU, March 4 (Reuters) - Oil rallied more than 4% on Thursday, hitting its highest in over a year, after OPEC and its allies agreed to keep production unchanged into April, reasoning that the demand recovery from the coronavirus pandemic was still fragile.

Brent crude rose $2.67, or 4.2%, to settle at $66.74 a barrel, after rising to $67.75, its highest since January 2020.

U.S. crude futures ended $2.55, or 4.2%, higher at $63.83, having also scaled a January 2020 peak, at $64.86.

"OPEC surprised us... The message OPEC is sending market is they're quite willing to see oil prices run hot and ultimately, go a long way in reducing the inventory overhang built last year because of COVID-19," said Bart Melek, head of commodity strategies at TD Securities.

Some analysts had predicted OPEC+, an alliance of the Organization of the Petroleum Exporting Countries and other major producers, would increase output by about 500,000 bpd.

The group's leader Saudi Arabia said it would extend its voluntary oil output cut of 1 million barrels per day (bpd), and decide in coming months when to gradually phase it out.

"There is one thorn to the bullish cocktail though and very few are surprised. Russia wants to boost output," head of oil markets at Rystad Energy, Bjornar Tonhaugen said in a note.

Russia was allowed to raise output by 130,000 bpd in April and Kazakhstan by another 20,000 bpd.

"Russia aside, the biggest winner of an OPEC+ rollover is the U.S.. With such price levels, which are now boosted even more after the news of a possible rollover consensus, the U.S. can comfortably increase production, even from costly break-even projects," Tonhaugen added.

Also supporting sentiment, Yemen's Houthi forces said they had fired a missile at a Saudi Aramco facility in Jeddah.

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(Reporting by Arpan Varghese and Sumita Layek in Bengaluru; Additional reporting by Bharat Govind Gautam, Naveen Thukral and Florence Tan Editing by Marguerita Choy and David Gregorio)

((Shadia.Nasralla@thomsonreuters.com; +44 207 542 5083; +44 778 99 43141; Reuters Messaging: Reuters Messaging: shadia.nasralla.reuters.com@reuters.net))