* U.S. crude stockpiles fall 4.1 mln bbls last week- EIA
* Estimated U.S. gasoline demand hits record high 9.9 mln bpd -EIA
* Oil demand steady in 2019 but OPEC supply, trade risks loom-IEA
* Trump, Iran spar over oil prices ahead of OPEC meeting (Adds settlement prices, Fed move)
NEW YORK, June 13 (Reuters) - Oil prices turned positive on Wednesday after a bigger-than-expected decline in U.S. crude inventories along with surprise drawdowns in gasoline and distillates indicated strong demand in the world's top oil consumer.
Earlier in the session, Brent and U.S. crude futures had retreated on concerns about rising production in the United States and expectations that OPEC and other producers could relax voluntary output cuts when they meet on June 22-23 in Vienna.
Brent crude settled up 86 cents, or 1.1 percent, at $76.74 a barrel and U.S. crude closed 28 cents, 0.4 percent, higher at $66.64 a barrel.
Late in the session, crude prices slipped slightly as the Federal Reserve raised interest rates, a move that was widely expected but still marked a milestone in the U.S. central bank's shift from policies used to battle the 2007-2009 financial crisis and recession. Higher interest rates strengthen the dollar, increasing the cost of commodities including oil for buyers using other currencies.
U.S. crude inventories fell 4.1 million barrels last week, the Energy Information Administration said, exceeding analysts' expectations for a decrease of 2.7 million barrels. Estimated U.S. gasoline demand hit a record high of 9.9 million barrels per day (bpd) in the week, the data showed.
"The demand metrics here are amazing for crude oil and gasoline," said John Kilduff, a partner at Again Capital in New York. "Put the exports of crude on top of that, and it's just a really bullish report."
U.S. crude production rose to 10.9 million bpd last week, according to the EIA, but Kilduff said the market appeared able to absorb the increase. "It seems like we need almost every barrel of that to keep up with this refining demand."
With output in Russia rising back above 11 million bpd in June and Saudi production climbing to more than 10 million bpd, supplies from all three top producers are increasing.
The Organization of the Petroleum Exporting Countries and some non-OPEC producers, including Russia, started pumping less in 2017 to reduce a global crude glut. Prices have risen around 60 percent over the last year.
"More oil from OPEC plus is the base case," said Bjarne Schieldrop, analyst at Swedish bank SEB.
"Saudi Arabia and Russia have already started to lift production," he said. "Unofficial sources have said Russia will propose to return production back to the October 2016 (level), i.e. removing the cap altogether over a period of three months."
U.S. President Donald Trump and Iran exchanged sharp words over oil prices, with Trump blaming OPEC for high oil prices and Tehran accusing him of stoking volatility after he withdrew last month from a global nuclear arms deal with Iran.
Longer term, the market could tighten as demand increases if OPEC fails to cover supply shortfalls, the International Energy Agency said.
The IEA said it expects global oil demand to grow 1.4 million bpd this year, and in 2019, and will top 100 million bpd in the fourth quarter of 2018.
"The market will be finely balanced next year, and vulnerable to prices rising higher in the event of further disruption," the IEA said in its monthly report.
Fund manager Pierre Andurand at Andurand Capital was bullish.
"Prices will be above $150 in less than two years," he tweeted.
(Additional reporting by Henning Gloystein in Singapore and Christopher Johnson in London; Editing by David Gregorio and Marguerita Choy)