The goods, worth almost 20,000, were found on Tuesday afternoon after a tip-off that a man was storing illicit cigarettes in a van near his home.
Customs officers found almost 10,000 cigarettes in the van, then raided an address in Brunswick Road where a further 52,000 cigarettes and 35kg of tobacco were found.
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Hide AdSergeant Andy Johnson said: "It was discovered through a combination of police intelligence and intelligence that customs received through their hotline. This is evidence that we do act on local intelligence received and get positive results."
A 51-year-old man has been cautioned as part of the enquiry, but no arrests have been made.
An HM Revenue & Customs spokesperson said: "The goods are non-UK duty paid, equating to around 14,000 in evaded duty. There were no arrests and enquiries are ongoing.
"Anyone with information regarding suspected illegal tobacco or alcohol products in the Leith area can call the HMRC hotline on 0800 59 5000."