I suspect that this is one of those things that was essentially well-known to many people before anybody wrote it down, so it will be hard to pin down the first person to prove it.

But the main idea goes back to

*Happel, Dieter*, **On the derived category of a finite-dimensional algebra**, Comment. Math. Helv. 62, 339-389 (1987). ZBL0626.16008.

There Happel considered the bilinear form on $K_0\left(D^b(\text{mod }A)\right)$, for an algebra $A$ of finite global dimension, given by $\left\langle[X],[Y]\right\rangle=\sum_i(-1)^i\dim\text{Hom}(X,Y[i])$. To get the statement about the Cartan determinant just requires the idea of doing the same for $K_0\left(K^b(P_A)\right)$ for an arbitrary finite dimensional algebra, where $K^b(P_A)$ is the triangulated subcategory of $D^b(\text{mod }A)$ consisting of bounded complexes of finitely generated projective modules.

The derived invariance of the Cartan determinant is explicitly stated as Proposition 1.5 in

*Bocian, Rafał; Skowroński, Andrzej*, **Weakly symmetric algebras of Euclidean type.**, J. Reine Angew. Math. 580, 157-199 (2005). ZBL1099.16002,

where they credit me with suggesting the proof. But I don't remember doing so, and I suspect it may have been in a conversation at some conference many, many years earlier. I also suspect that I was not the only person at the conference who would have suggested exactly the same proof.