This represents a fourfold rise on the total last year.
Most crossed by sea, with more than 800,000 travelling from Turkey to Greece. Half are migrants from Syria.
Eleven more migrants drowned on Tuesday, adding to the IOM toll of 3,695 dying or missing at sea.
Seven people were rescued by Turkish coast guards after the craft went down, apparently en route from Kusadasi in Turkey to the Greek island of Samos. One report said the dead were Syrians.
Migrant crisis in graphics
Shelter or despair in Athens
Desperate journeys
The huge influx of migrants has caused significant political rifts within the EU, with some states inside the border-free Schengen area putting up fences and reimposing frontier controls.
Hungary and Slovakia are taking legal action at the European Court of Justice to challenge EU plans to share asylum seekers across EU states.
Meanwhile, many migrants and refugees are pressing to be allowed to settle in richer northern countries like Germany and Sweden.
Migration passed the symbolic milestone on Monday, the IOM said, with the total for land and sea reaching more than 1,006,000.
Entries via six EU nations - Greece, Bulgaria, Italy, Spain, Malta and Cyprus - are covered in the report.
It found among other things that
The IOM gathers its statistics from registrations, law enforcement agencies and its own monitors.
Its director general, William Lacy Swing, said it was not enough to just count the figures.
"We must also act," he said. "Migration must be legal, safe and secure for all - both for the migrants themselves and the countries that will become their new home."
A joint IOM and UNHCR statement said found a "more co-ordinated European response" was beginning to take shape.
However, it said more needed to be done to improve reception facilities, accommodation and  registration, and to identify those who do and do not qualify for refugee protection.
Save the Children campaigns director Kirsty McNeill said: "This is the test of our European ideal. When children are dying on our doorstep we need to take bolder action. There can be no bigger priority."
The EU last week agreed to increase the numbers of Frontex border agency staff in Greece, a key arrival point.
Germany alone has received a million refugees and migrants this year, although many were already within Europe, particularly in the Balkans.
Macedonia is now refusing to allow anyone through its Greek border who does not come from a war zone.
A UN report also last week warned that the number of forcibly displaced people worldwide would "far surpass" 60 million this year.
A note on terminology: The BBC uses the term migrant to refer to all people on the move who have yet to complete the legal process of claiming asylum. This group includes people fleeing war-torn countries such as Syria, who are likely to be granted refugee status, as well as people who are seeking jobs and better lives, who governments are likely to rule are economic migrants.
The number of migrants and refugees crossing into Europe by land and sea this year illegally has passed one million, the International Organization for Migration (IOM) says.