Let $A, B$ be dg Lie algebras. I'm trying to prove that if $f: A \rightarrow B$ is a quasi-isomorphism then $\otimes^n f: A^{\otimes n} \rightarrow B^{\otimes n}$ is also a quasi-isomorphism. I'm willing to assume that we are working over a field $K$ of characteristic zero, and that $A$ and $B$ are bounded (only nonvanishing positive degrees), if we need to. I think it is enough to prove it for the case $n = 2$, and for that I was trying to show that the mapping cone of $f \otimes f$ is acyclic using the fact that $f$ is a quasi-iso, but I haven't managed to conclude anything from it yet.
I'm new to homological algebra so I think I might be missing some key fact that would make concluding this proof much easier, so I would appreciate if someone could nudge me in the right direction (maybe some reference). Thank you!