The war over oil market is just beginning. And in its first battle against OPEC, the U.S. appears to have blinked.

The U.S. crude output growth buckled last month, as the rig count continues to fall and producers face pressure to cut costs, the International Energy Agency said Wednesday. The output slipped 25,000 barrels a day to 12.6 million barrels a day in April.

Meanwhile, OPEC has boosted its production in an attempt to squeeze other emerging players and protect its market share, the Paris based IEA said.

But the IEA, which monitors energy market trends for the world's richest nations, said it would be "premature" to suggest the Saudi-led cartel has won the battle for market share.

"The battle, rather, has just started," the report said.

Related: End of OPEC is closer to reality

Other non-OPEC countries defied the price pressure and ramped up production in April, the agency's data show.

Brazil's output was up 17% year-on-year in the first quarter, while Russian producers managed to overcome western sanctions and low prices, producing 10.7 million barrels per day in April, 185,000 barrels a day more than in the same period last year.

OPEC crude supply rose by 160,000 barrel per day to 31.2 million barrels a day in April -- the highest since September 2012.

Although Saudi Arabia trimmed its output by to 10.1 million barrels a day in April, it maintained its production above the 10 million barrels a day mark -- clearly showing the determination to preserve its market share, the report said.

April was also the 12 consecutive month in which production ran above OPEC's 30 million barrels a day supply target.

The Organization of the Petroleum Exporting Countries agreed in November not to cut production despite the collapsing prices. Oil prices dropped by about 50% in the last six months of 2014 but have stabilized since then.

And the IEA said that was just the first step in a plan to keep its market share.

"(OPEC countries are) aggressively investing in future production capacity -- even as their non-OPEC counterparts keep tightening their belt," the report said.

Oil was up 0.8% at $61.3 per barrel on Wednesday morning.