Adds details on crude inventories and exports in paragraph 3, 6, 7, prices in paragraph 8, refined product demand comments in penultimate paragraph

By Arathy Somasekhar and Liz Hampton

HOUSTON, June 3 (Reuters) - U.S. crude stocks fell on strong export and refining demand as the U.S.-Israel war on Iran entered its fourth month, while gasoline and distillate inventories rose last week, the Energy Information Administration said on Wednesday.

Crude inventories fell by 8 million barrels to 433.7 million barrels in the week ended May 29, the EIA said, compared with analysts' expectations in a Reuters poll for a 4 million-barrel draw.

Stocks along the Gulf Coast drew down by 6.7 million barrels as oil exports touched 5.9 million barrels per day, the second highest level on record.

Net U.S. crude imports USOICI=ECI fell last week by 249,000 barrels per day, EIA said.

Crude stocks at the key storage hub in Cushing, Oklahoma USOICC=ECI fell by 583,000 barrels in the week, the EIA said.

Overall, crude inventories have declined by 63.9 million barrels since the Iran war began on Feb 28.

"Another large draw in US crude inventories, driven by both a drop in commercial and strategic inventories," said Giovanni Staunovo, analyst with UBS.

Oil futures extended gains following the report. Global Brent crude futures LCOc1 were trading at $97.52 a barrel, up $1.52 at 10:37 a.m. ET (1436 GMT), while U.S. West Texas Intermediate futures CLc1 were up $1.58 to $95.34 a barrel.

Refinery crude runs USOICR=ECI fell by 90,000 barrels per day, the EIA said. Refinery utilization rates USOIRU=ECI rose by 0.2 percentage points to 94.7% in the week.

U.S. gasoline stocks USOILG=ECI rose by 3.4 million barrels in the week to 215 million barrels, the EIA said, compared with analysts' expectations in a Reuters poll for a 0.5 million-barrel draw.​

Distillate stockpiles USOILD=ECI, which include diesel and heating oil, rose by 1.5 million barrels in the week to 102.3 million barrels, versus expectations for a 0.3 million-barrel drop, the EIA data showed.

"The refined product side looks less bullish, with surprising builds for gasoline, distillate, and jet fuel, due to lower implied demand post the Memorial Day holiday," said Staunovo, adding that demand for products should increase in coming weeks.

Total product supplied, a proxy for demand, fell by 610,000 barrels per day to 20.94 million bpd.