Paradoxically, having the Soviet military control Kremlin policy will not necessarily make Soviet policy more militaristic. Not all generals are gung-ho warriors; most will want their forces to be strong as possible and many might desire to exercise their forces in sure-thing operations, but many, particularly Soviet military leaders who survived the "Great Patriotic War," are also keenly aware of the costs and dangers of reckless actions and, even if their own are in charge of government policy, might shrewdly fear that in any major war they might wind up out on a limb with the regime unable or unwilling to give them the support they need to follow through to victory. A series of events that puts the Red Army leadership in charge in Moscow might well lead to a period of cautious bunker mentality and feelers going out to the West for detente and a modus viviendi.
It's pretty hard to see how the Soviets could manage to get ahold of even one of the major Middle Eastern oil-producing states without the West rearing up and slapping them down somehow. Much less unlikely (though still pretty out there) than getting control of Arabia would be to gain strong hegemony over Iran, which would at least put Soviet power on the Persian Gulf coast. During WWII northern Iran was practically a Soviet protectorate, but southern Iran "enjoyed" the same sort of "protection" by Britain--to parley his wartime strength into lasting domination all the way to the southern coasts (which, if I am not mistaken, is where the oil is) would require amazing footwork on Stalin's part, surely must involve raising up and then co-opting a very strong domestic Iranian Communist movement, one that could outmaneuver the best anti-Communist tactics the British and Americans combined could muster. It isn't clear to me that the post-war USA would simply threaten nuclear war if the Russians didn't back out of Iran regardless of domestic conditions there, but it isn't clear to me we wouldn't either.
To go farther, to secure control of Iraq (not so unthinkable if they already have Iran, but it's that first step that the killer) and then on to Saudi Arabia, means that every step of the way, the Westerners see the Soviets gain and themselves lose, and their resolve to stop it and if possible reverse it must redouble with every loss they suffer.
Can anyone come up with a plausible scenario whereby Soviet power leapfrogs direct to Saudi Arabia? I sure can't! There was, in the timeframe of WWII when such steps would be least unthinkable, hardly any industrial development in Arabia at all; the very sparse population (not, then, that difficult to overpower with big Western forces if it came to that, and not capable of easily supporting large Soviet forces however welcome they might be) was about as far from a natural ally of Soviet rule as anyone might imagine, being both pre-capitalist and highly religious. They may or may not have been winnable to some anti-Saudi venture, but not I think to enlisting under the Communist banner. (This is a major reason US policy has on the whole supported the Saudi regime so strongly).
So even if the Russians were successful and unstoppable in gaining control of Iran and Iraq, and therefore poised to possibly try to march into Saudi Arabia by main force, they'd be taking on the most difficult people imaginable to conquer, and the Arabians would surely have the Western nations come a-running to defend them at all costs, right up to nuclear brinksmanship and if necessary, Armageddon itself. Until the later 1960s, the Soviet Union's ability to inflict serious harm on North America at any rate was in great doubt; much caution in the West was due to a prudent (sometimes, not so much prudent as demogougic!) worst-case assumption reinforced by Soviet bluster, so before 1965 or so it wouldn't be unthinkable for Americans to follow through on nuclear threats if the Soviets did something as outrageous as actually invading Arabia.
So, Soviet Arabia requires a POD well before 1945, one that makes indigenous Communism in the Arabian peninsula much stronger, so strong that Arab Communists could take over on their own with minimal to no help from Russia--even if the Russians could spare material and manpower aid, they'd have a hard time logistically getting it there past a gauntlet of European-colonial controlled territories! If this happens well before 1939, when Arab oil was as yet largely undeveloped, it could be that the colonial powers of the region (ie, entirely actually, Britain) might judge the cause lost and refrain from intervention--but that hardly seems likely. Nor is it clear how the Russians would get help to the Arab Communists. Unless of course they had after all first got control of Iran by similar means, and used Iran as the conduit to send aid southward.
So you see, if you want the Soviets to take one Middle Eastern region, best it be Iran. And I don't believe they'd be allowed to go any farther than that, for that matter taking Iran would meet with the utmost Western resistance possible. Iraq might be hard for Westerners to hold against a strong Soviet Iran, but that would be the limit--and they'd fight hard to hold Iraq.