As many as six suicide car bombs and mortars were used in the assault on the compound that houses the main police HQ and governor's office.
At least 50 police officers are reported to have been taken prisoner at the site.
IS and Iraqi troops have been battling for months to take control of the strategically important Anbar province.
This latest attack comes a day after Islamic State put out an audio message it claimed was from its leader, Abu Bakr al-Baghdadi,who Iraq had said was seriously wounded in a coalition air strike in March.
Execution fears
IS militants launched their raid on Ramadi overnight, driving suicide car bombs into the compound housing a number of government buildings.
At least 10 police officers were killed and dozens of others wounded in the attack, officials said.
Fighting continued into Friday and by 14:00 (11:00 GMT) the black flag of IS was seen flying over the complex.
IS "now occupies the government centre in Ramadi and has also raised its flag over the police HQ for Anbar", a police major told the AFP news agency.
The militant group itself issued a statement confirming it had taken control of the complex and said it had killed an unspecified number of pro-government fighters.
Fifty police officers are known to have been taken prisoner in the assault, but reports that they have been summarily executed are unconfirmed.
Analysis: Ahmed Maher, BBC News, Baghdad
Islamic State wants to send a clear message to the central government in Baghdad that, having seized control of Ramadi, they are now the de facto rulers of Anbar province.
Supported by anti-government Sunni tribes and members of former President Saddam Hussein's army, the militant group has been trying for months to seize Ramadi.
And, after several false starts, they have succeeded in taking control of the city centre and the main government complex. This is an extraordinary victory that will give IS the momentum to control the whole of Anbar.
Since last year, the militants have made big territorial gains in Anbar and in the northern province of Nineveh, which is home to the group's stronghold city of Mosul.
This latest assault is a blow for the Iraqi government which has been trying for more than a year to prevent Anbar and its key towns and cities from falling into the hands of IS.
The heavily-Sunni province of Anbar covers a vast stretch of the country west from the capital Baghdad to the Syrian border, and contains key highways that link Iraq to both Syria and Jordan.
Iraq's Prime Minister Haider al-Abadi pledged in April that his forces would "liberate" Anbar from IS after the success of re-taking Tikrit.
However, by Friday, the militant group had asserted its control over large areas of Ramadi and half of Anbar province, our correspondent reports.
Meanwhile, in Syria, Islamic State militants are said to be getting close to the famous ancient ruins of Palmyra.
Syrian government forces and warplanes are trying to halt the advance of IS fighters towards the World Heritage Site.
Cultural officials fear that if IS makes it to Palmyra, they will destroy the historic ruins like they have at several other ancient sites in Iraq.
Troubled history of Anbar province
- Iraq's largest province and its only Sunni-dominated one was occupied by US forces in 2003
- Hostile to the US, fighting quickly broke out between US troops and the region's Sunni insurgents
- The worst battle came in 2004 when thousands died as US troops and coalition forces struggled to take the town of Falluja
- Fighting continued in 2005 and 2006 during which time al-Qaeda in Iraq (AQI) rose to prominence
- The US declared victory in 2007 but AQI remained, resuming attacks in 2011 when US troops withdrew
- AQI, which appears to have now become the Islamic State, and other Sunni insurgents currently control much of the province
