Brent crude futures were last down 23 cents on the day at $78.83 a barrel by 3pm, having touched a session peak of $79.66, the highest since late May, when the price broke above $80.
USA sanctions on Iran's energy industry, which come into force in November, have already cut supply back to two-year lows, while falling Venezuelan output and unplanned outages elsewhere will also keep the balance between supply and demand tight, the IEA said.
"We think oil market fundamentals are increasingly supportive of crude prices, at least at current levels", said Gordon Gray, HSBC's global head of oil and gas equity research.
United States crude futures were up 35 cents at $69.60 a barrel.
Prices extended gains in post-settlement trade after industry data from the American Petroleum Institute showed USA crude inventories slumped 8.6 million barrels last week, versus analysts' forecasts of a 805,000-barrel decrease. "The situation in Venezuela could deteriorate even faster, strife could return to Libya and the 53 days to November 4 will reveal more decisions taken by countries and companies with respect to Iranian oil purchases", the IEA said, referring to the day USA sanctions on Iran take effect.
But the U.S. government does not want to push up oil prices, which could depress economic activity or even trigger a slowdown in global growth.
U.S. Energy Secretary Rick Perry met Saudi Energy Minister Khalid al-Falih on Monday in Washington, as the Trump administration encourages big oil-producing countries to keep output high. Perry will meet with Russian Energy Minister Alexander Novak on Thursday in Moscow.
"Crude oil output increased mostly in Libya, Iraq, Nigeria and Saudi Arabia, while production declined in Iran, Venezuela and Algeria", the 15-member oil cartel said.
US crude inventories fell 5.3 million barrels in the week to September 7 to 396.2 million barrels, the lowest since February 2015 and about 3 per cent below the five-year average for this time of year, the US Energy Information Administration (EIA) said on Wednesday.
Short-term, the outlook is for tighter supply.
The Organisation of Petroleum Exporting Countries, in its monthly oil market report for September released on Wednesday, said crude oil production from Nigeria rose to 1.71 million barrels per day in August from 1.53 million bpd the previous month.
At the same time, American oil producers are seeking new buyers for crude they used to sell to China before orders virtually dried up because of the trade disputes between Washington and Beijing.