In a rapid risk assessment, the European Centre for Disease Prevention and Control (ECDC) said that 25 cases of the disease - which is also called "breakbone fever" because of the severe pain it can cause - have been found elsewhere in Europe in travellers returning from Madeira.
Such cases have so far been picked up in Portugal, Britain, Germany, Sweden and France, it said.
"Given the dramatic expansion of endemic dengue transmission globally over the last 20 to 30 years and the high number of visitors to Madeira, the outbreak is large and constitutes a significant public health event," the ECDC said in an assessment issued late on Tuesday.
Since the outbreak began in early October, 1,357 cases of dengue fever have been reported by health workers in Madeira, including 669 laboratory-confirmed cases and 688 probable cases.
Eighty-nine people have received hospital treatment but there have been no deaths so far.
Dengue is a viral infection that can cause a range of symptoms, from mild flu-like illness to more serious illnesses including rashes and bone pain. Severe and potentially deadly forms develop in around 5 percent of patients.
The most efficient carriers of the disease, mosquitoes known as Aedes aegypti, have an established presence in Madeira, an archipelago north of the Canary Islands in the Atlantic Ocean.
The ECDC, which monitors disease in the region, said it was working with European Union member states and the European Commission to make sure cases in returning tourists were fully reported and tracked.
Local transmissions of dengue fever in Europe were recorded in France and Croatia in 2010, but the Madeira outbreak is the first sustained transmission of the disease since the 1920s and is expected to continue to the end of the year.
Beyond that, "mosquito density will probably decrease", the ECDC said.
Earlier this year, Greek health officials attributed the death of an 80-year-old man to its first case of dengue since an outbreak there in 1927-28. Greece is suffering an upsurge in mosquito-borne diseases, including malaria.
The ECDC did not recommend any Madeira travel restrictions, but advised protection against mosquito bites, including disinfection and control of mosquitoes at ports and airports.
Since dengue-carrying mosquitoes are daytime biters, it said protection should be applied throughout the day.
The ECDC also advised authorities in geographical areas neighbouring Madeira, such as the Canary Islands, to step up surveillance of Aedes mosquito populations to assess the risk that dengue fever might spread.