Functorial cones

The cone construction can be written down very explicitly, just following the definition of mapping cone of chain complexes. Good sources are in my opinion:

https://arxiv.org/pdf/math/0401009.pdf Definition 3.7

https://arxiv.org/pdf/math/0210114.pdf paragraph 2.9, where it is discussed how the cone is functorial as a dg-functor from the (homotopy coherent) dg-category of morphisms.

Or, if you like, I wrote both a master and a PhD thesis centered on dg-categories: https://anisama.files.wordpress.com/2019/04/tesi_mag.pdf (master thesis) https://anisama.files.wordpress.com/2019/04/tesi.pdf (phd thesis)

A proof in the context of model categories can be found in Proposition 6.3.5 of Hovey's book Model Categories. You could easily rewrite the proof to work in the context of dg-categories, where it is actually easier.

EDIT: Here is a source that does it for dg-categories, in Section 4.1