Prime Minister Haider al-Abadi said the battles marked the end of the war – three years after IS militants seized a third of the country’s territory.
Last week Russia said that its forces had destroyed all IS strongholds in Syria following a bombing campaign.
Colonel-General Sergei Rudskoi, head of the general staff’s operations, said: “The mission to defeat bandit units of the Islamic State terrorist organisation on the territory of Syria has been accomplished.”
Iraq and Syria were the axis of the IS “caliphate”.
Western security experts are now trying to determine where battle-hardened IS leaders have gone and whether they could be heading for Britain.
Theresa May said: “On behalf of the United Kingdom, I congratulate Prime Minister Abadi and all Iraqis on this historic moment. I pay tribute to the Iraqi security forces for their courage.”
One IS leader was arrested last week in Tangiers in Morocco and three Moroccans have been held in Spain on suspicion of plotting terrorist attacks.
Two Moroccan brothers, aged 30 and 31, were detained last week at Figueres on the Costa Brava, near the frontier with France.
They were alleged to have been running a complex communications network to generate jihadist propaganda.
A 44-year-old Moroccan with Spanish nationality was arrested at Parla, near Madrid, by counter intelligence officers who believe he was seeking accomplices for an attack.
While Europe prepared for a backlash, leaders in Iraq and Syria were jubilant about their success.
Mr al-Abadi said: “Our heroic armed forces have now secured the entire length of the Iraq-Syria border. Our enemy wanted to kill our civilisation, but we have won through our unity and our determination. We have triumphed in little time.”
Following the announcement that IS had been driven out, Iraq’s oil minister announced an export contract with neighbouring Iran after years of conflict between the two countries.
The writing on the wall for IS came last month when Iraqi forces captured Rawa, the last town under IS control.
Mosul, IS’s de facto capital in Iraq, fell in July after a nine-month battle which left most of the northern city destroyed.
In September IS’s Syrian capital Raqqa fell to a US-backed Kurdish-led coalition.
IS fighters flushed out of Iraq are reported to have dispersed into the Syrian countryside, while others are believed to have escaped across the Turkish border.
A global intelligence effort is under way to try to track them to see whether they are disbanding or forming small militias, which could still be capable of inflicting devastating attacks.
RAF Commodore Johnny Stringer had warned earlier this year that IS would “almost certainly morph into an insurgent organisation”.
Islamic State’s cross-border province, or caliphate, was a symbol of the jihadists’ intention to eradicate all the region’s frontiers and lay to rest the 1916 Sykes-Picot agreement, an emblem of the colonial division of the area which was resented by some Arabs and Kurds.
Today that ambition lies in tatters, but no security chiefs believe that the world has heard the last of them and fear random attacks will continue for years to come.