For a short time Iran naturally gained some points in Iraq among the Shias because Saddam Hussein oppressed Shias in Iraq, and Shia (but Persian) Iran also bitterly opposed Saddam. And while post-Saddam Iraq has been weak, Iran has been able to project power and influence into Iran.
But that is changing:
Iraq, which Iran hoped to annex or at least turn into a compliant client state, is not cooperating. In part because Arab Shia have always disliked the Iranian (Indo-European) Shia and vice versa. Iraqi Shia get regular reports of how badly Iran still (after thousands of years) treats its Arab minority. So while the Shia dominated government of Iraq pretends to be friendly and grateful towards Iran it is becoming obvious to Iran that this is all for show and not real. Opinion polls in Iraq make it clear that most Iraqis are hostile to Iran and Iranian intentions. These suspicious attitudes are on the rise. This does not mean the Iraqis are above making some cash in somewhat questionable transactions. But bending the knee to Iran is not on the menu.
I suppose in 1946 with the same logic that the Army might have concluded that the only party to have won World War II was the USSR.
Although to be fair, if we'd left Europe soon after 1945 we might have gotten the same kind of defeat in Europe that we faced in Iraq after we left too soon in 2011. Although defeating a major military power would have been a far harder war than defeating the ISIL proto-state in Iraq and Syria at our leisure.
With our enduring presence in Iraq following the defeat of ISIL, we will be able to help Iraq push Iranian influence out of Iraq. And our presence in Iraq will finally get Arab states to work with Iraq in pan-Arab solidarity rather than let Sunni-Shia divisions keep them apart.
UPDATE: Iraqis appreciate the need for American troops in Iraq to watch Iran--and Turkey and Saudi Arabia:
Most Iraqis understand that without the presence of those Americans Iraqi would be even more threatened by most of their neighbors (Iran, Turkey and Saudi Arabia). Even many Iraqis who are not particularly pro-American recognize this use of American troops as a “protection force.” The Americans aren’t there to fight Iraqis but to keep the neighbors from making life difficult for an independent Iraq.
And hopefully we watch the Iraqi government and military to make sure they don't deteriorate the way they did between our 2011 departure and the 2014 ISIL uprising that collapsed large parts of the security forces. We should be able to spot problems and advise the Iraqis in a timely manner.
That presence to prevent the rot from spreading again is necessary because ISIL survives underground and could regenerate if Iranian pressure succeeds in alienating Iraqi Sunni Arabs again.