There's a nice, short proof in

*Positselski, Leonid; Schnürer, Olaf M.*, **Unbounded derived categories of small and big modules: is the natural functor fully faithful?**, J. Pure Appl. Algebra 225, No. 11, Article ID 106722, 23 p. (2021). ZBL1464.18015.

where this is Proposition 4.1(b).

Suppose $R$ has global dimension $d<\infty$. Let $P^\bullet$ be a complex of projectives, and let $\alpha:P_K^\bullet\to P^\bullet$ be a $K$-projective resolution. To prove that $P^\bullet$ is $K$-projective, it suffices to prove that $\alpha$ is a homotopy equivalence, or equivalently that the mapping cone of $\alpha$ is contractible.

So we just need to prove that an acyclic complex of projectives is contractible.

Let $Q^\bullet$ be an acyclic complex of projectives. Then the truncation
$$\dots\to Q^{-2}\to Q^{-1}\to Q^0\to 0\to\dots$$
is a projective resolution of some module, which has projective dimension at most $d$ by the assumption on the global dimension of $R$. Therefore the image of the differential $Q^{-d}\to Q^{-d+1}$ is projective.

Applying the same argument to shifts of $Q^\bullet$, every differential of $Q^\bullet$ has projective image, which (together with the fact that $Q^\bullet$ is acyclic) implies that $Q^\bullet$ is contractible.