But there was some oil production in Persia, and they had struck oil in Iraq, Saudi Arabia and Bahrain in the 1930s. ARAMCO was established in 1943.
Oil exploration in Persia / Iran dated back to 1901 and was discovered in 1908 leading to exploitation by the Anglo Persian Oil Company (later Anglo Iranian Oil Co later BP). By WW2 their refinery at Abadan was I believe the single largest in the world. It became the biggest source of oil for the Admiralty in WW1. Exploration from there then spread around the Gulf through a network of British companies. The US got involved in WW2 to help increase production and refining capacity in the region.
Iraq oil developed interwar following the collapse of the Ottoman Empire and the acquisition of German oil concessions by British & French interests. Oil was piped across the desert from 1934 to refineries at Haifa in Palestine (then a British mandate) and Tripoli in Lebanon (then a French mandate).
There were also some small oilfields in Egypt along the Red Sea Coast by WW2. But exploration in the Western Desert had to wait until the 1970s to be discovered.