this question was actually asked recently, and answered with a nice quote from The Lives of the Artists (see link below). Although Vasari was alive in the 15th century, this awareness is evident as early as the works of Petrarch in the 14th century.
The use of the term Renaissance has clouded historiography since then and created an image of the period from the Fall of the Roman Empire to the rise of Petrarch and his contemporaries as a Middle Period of stagnation that took many years to demystify.