The earliest available evidence of 'Africa' applied to the whole continent (including Egypt) would appear to date from the work of 16th century map makers. Abraham Ortelius (1527-98) produced this map in 1584:
Source: Evolution of the Map of Africa
A map by Rumold Mercator comes just 3 years after Ortelius' but judging by the colouring, Madagascar is clearly excluded. This map was presumably primarily the work of his more famous father Gerardus Mercator whose first world map dates back to 1538, but there I've been unable to find a clear reference to Africa applied to the whole continent.
Source: Slika:Mercator World Map
An editor's note in the General History of Africa, vol. 1 says:
From designating the North African coast, the word 'Africa' came to be
applied to the whole continent from the end of the first century
before our era.
True enough if one excludes the area east of the Nile (see Semaphore and fdb's answers), but the Romans and the Byzantines had Egypt as a province, separate from (North) Africa. The Arab cartographer Muhammad al-Idrisi (1100-65) made a number of maps but does not seem to show Egypt as part of the African continent (there's a pdf download here: World Maps of al-Idrisi). Al-Idrisi, who was in the court of Roger II of Sicily, travelled widely and produced a work titled (Wikipedia translation) Recreation of the desirer in the account of cities, regions, countries, islands, towns, and distant lands. This may possibly contain a reference to 'Africa' for the whole continent, but I've been unable to find a copy.
In the 14th century, included in the document World Maps of al-Idrisi is a later (1381) world map in "Ibn Khaldun’s monumental work, The History of the World, 1381", but this too provides no evidence.
Ortelius's map was followed by other maps showing Egypt included in the African continent (see this 1656 one and this one from 1710).