The war ended with a return to the status quo ante bellum - back to the pre-war borders. Neither side achieved what they wanted, and both countries suffered badly from the war. Neither side gained, so how could either be called a "winner"?
It's a little more complicated than that. Iraq failed to achieve any of its original war aims, but it did achieve its final war aims. Iran, on the other hand, achieved its initial war aims, but then expanded its goals a lot, and failed to achieve those expanded war aims.
Iraq started the war with three main goals. Their minimum goal was to restore the Iran-Iraq border along the Shatt al-Arab waterway (connecting the Tigris-Euphrates to the Persian Gulf) back to the 1937 border (along the east bank of the river, making the river itself Iraqi). The 1937 border was disputed from 1969 to 1975, when a new treaty set the border mid-channel (as it had been before 1937). Iraq saw this as Iranian bullying and a national humiliation. Thus, restoring the 1937 east-bank border was an important war aim for Iraq.
Iraq also wanted control of Arab-majority oil-rich Iranian province of Khuzestan, bordering Iraq at the northern end of the Persian Gulf. Iraq denied wanting to annex Khuzestan, but at the least, they wanted control of it as an "independent" puppet state carved off from the rest of Iran.
Their third major war aim was to overthrow the Islamic revolutionary government in Iran (which Iranian exiles claimed would fall due to the invasion).
The Iranian war aims in response to the invasion were, in order of importance, to preserve the Islamic revolution and to protect Iranian territory. If necessary, territory would have been sacrificed to preserve the revolution. As it was, the Iraqi forces only made significant advances for a few months, and pushing Iraq out of Iran was a feasible war aim. In the end, Iran achieved these war aims. Indeed, the war strengthened the control of the revolutionary government over Iran.
Iranian success early in the war convinced the Iranian leadership that they could take the war to Iraq, and forcibly export the revolution. This led to the main part of the war (the last 3/4 of the war), a futile attempt to do this. Many of Iraq's neighbours viewed the possibility of an Iranian-backed Islamic government in Iraq as alarming, and very dangerous for their own future existence, and supported Iraq's defence against Iran with loans and grants. Iraqi troops were also willing to fight harder to defend Iraq than to attack Iran, and the Iraqi government invested in the defensive effort desperately. From the Iranian point-of-view, offensive operations where difficult when the enemy had superiority in artillery, armour, air power, and had prepared extensive fortifications.
Iraq's final war goals were simple: fight back hard enough to push the Iranians to negotiation and a peace treaty. With heavy use of chemical weapons, and active US participation in the war (e.g., US attacks on the Iranian navy, and destruction of Iranian oil facilities), and the lack of progress on the ground, Iran agreed to peace, and the war ended.
If we break the war into two chunks - the Iraqi offensive phase (the first 1/4 of the war) and the Iranian offensive phase (the last 3/4 of the war) - both parts can be considered defensive victories for Iran and Iraq respectively (the Iraqi defensive victory being vert hard-fought and very damaging to Iraq). However, looking at the war as a whole, neither side achieved their offensive goals, and "no winner" is fair. It could also be claimed that Iraq lost by more, and while there was no winner, Iraq was the loser. In particular, the war wrecked the Iraqi economy, which was the main cause for the invasion of Kuwait, which led to Iraq joining Iran as a pariah state, in turn leading to the US invasion of Iraq and the overthrow of Saddam Hussein.