Police said 16 teenagers had been found at the city’s airport or wandering the streets in the last 18 months – nearly double the number of unaccompanied overseas children in care.
The youngsters are understood to have been trafficked by gangs via Russia to work in cannabis farms and nail bars.
Labour councillor Gordon Munro said: “Half of the unaccompanied children come from one area, which is new and not our experience before.
“What flights are they coming in on? Where are they starting from? What local connections are bringing them to Edinburgh of all places?”
The 16 Vietnamese cases have increased the number of overseas children being cared for in the city to 33. Looking after the 16 youngsters could generate an additional annual care bill of up to £1 million.
Mr Munro has called for a “multi-agency approach” to tackle the problem. He said: “I think the UK Government has to be involved as well, if these children are being trafficked from abroad.”
Recent trafficking cases include swoops on more than 280 nail bars in Edinburgh, London and Cardiff last December. The raids led to 97 arrests. And last September, gangster Hong Chuong Dang, 44, was jailed for a year for trafficking migrants to work in his Las Vegas Nail Bar in Bathgate.
Kevin Hyland, the UK’s anti-slavery commissioner, last week called for tighter regulation of nail bars to tackle the exploitation of girls and women trafficked from Vietnam. His report found that traffickers charge between £10,000 and £33,000 to smuggle people from Vietnam to Britain.
More than half of the victims are children who most commonly end up in cannabis cultivation or nail bars. However, many are also sexually exploited, according to the report.
A 2014 National Crime Agency study found that nearly three-quarters of trafficking victims forced into criminal activities, including cannabis cultivation, were Vietnamese.
The latest official figures from the National Referral Mechanism say Vietnam was the joint top overseas country of origin for trafficked children to the UK last year.
Of the 227 under-18s from Vietnam found in the UK, more than half – 122 – were forced to work as slaves.
A report by anti-child trafficking charity ECPAT UK found that Vietnamese children were the most likely to go missing and return to traffickers once rescued.