Michael Kennedy said exports to the southern European country, which had been the second-biggest market for the whisky, honey and herb liqueur, are less than half of the 45,000 cases which would have sold there at the peak of the market.
However, that has been offset by growth in the United States plus encouraging performance in Latin America.
Mr Kennedy said: "Our financial year ends at the end of this month and we will grow the business.
"It will be a little bit at the value level and a bit ahead in profit terms as long as nothing drastic happens in the next few days.
"We are starting to see growth coming back in the US which is our biggest market and we are also doing a lot in emerging economies in Latin America in Chile and Brazil.
"Southern Europe is a big issue. Greece was actually our second- biggest market and we have taken a bit of a hiding out there.
"We have seen our sales pretty much halve in Greece over the past three years.
"It is anybody's guess what will happen and whether Greece will stay in the euro."
Mr Kennedy outlined how Drambuie has tried to limit its bad-debt risk in Greece by running down stocks in the country.
He said: "It is almost one bottle in as one goes out. It is hard not to feel sorry for the Greek people as it is a pretty difficult situation."
Sales in the rest of southern Europe have also been affected.
Mr Kennedy said: "There is a real north-south divide in Europe for us.
"We are doing well in Germany, Holland and Scandinavia.
"Spain has suffered and Portugal, which was a really dynamic market with some very interesting growth, has flattened out."
Alongside Latin America the company will continue its focus on a select number of potential high growth markets – including India and Vietnam – in the coming year.
Mr Kennedy added: "We want to keep our growth markets list relatively tight. We could have 20 to 30 countries on that but we need to keep focused.
"Overall I'm happy with the way the business is going.
"The fact we are going to post a little bit of growth in the financial year is encouraging given the decline in Greece."
For the 2010/11 financial year Drambuie, which has offices at Newbridge near Edinburgh and in London, posted turnover of £21.9 million and underlying operating profits of £2.81m.