In a weird plea to American producers, all which are private enterprises as opposed to Opec members' state-owned oil companies, Barkindo said "extraordinary measures" might be required in 2018 to sustain the rebalanced market in the medium to long term.
Bullish forecasts from the International Monetary Fund also supported prices.
"A lot has been achieved towards stabilising the market, but to build on this success in 2018 will require continued discipline".
North American shale drillers have helped production soar by almost 10% in the USA this year, despite OPEC and some other producers, including Russian Federation, cutting supplies to prop up prices.
Barkindo said the effort in part helped the energy sector emerge from the "most vicious of all oil cycles", with oil prices tanking below $30 per barrel a year ago on oversupply concerns.
Together with its partners, which include Russia, Oman and Kazakhstan, the group has agreed to restrain output by 1.8 million bpd until March next year.
Prices rose 2 percent the day before to back above $50 a barrel.
"The next few weeks ahead of the producers' meeting in Vienna on 30 November will be crucial in shaping their decision on output", said the Paris-based IEA.
Barkindo's speech, delivered by video-link at the Reuters Global Commodities Summit in London, came after the secretary-general said Sunday that "extraordinary measures may have to be taken in order to restore" stability in the global oil market. The announcement appeared consistent with the kingdom's commitment to abide by an OPEC-led supply reduction agreement. United States crude inventories probably fell for a third straight week, while refined product stockpiles also likely declined, a preliminary Reuters poll showed on Tuesday.
"The IEA added ".it is this current outlook that might act as the ceiling for aspirations of higher oil prices".
Oil prices are now trading flat, after the Organisation of Petroleum Exporting Countries (OPEC) stated that the market is gradually tightening, after years of oversupply. However, US production is tipped to cap 10 million bpd in the second half of 2018, thanks largely to shale production, having risen by almost 10% in the year till date.
Bloomberg reported that the Brent benchmark had erased earlier declines to trade marginally higher at $55.62 a barrel, after the news of the Saudi oil allocations cuts. The main USA contract, WTI, was at $50.96.