Morphism of distinguished triangles where one of the arrows is a quasi-isomorphism

Let $R$ be any ring and let $Ato Bto Cto [1]$ and $A’to B’to C’to [1]$ be distinguished triangles of complexes of $R$-modules. Let $f:Ato A’$, $g:Bto B’$ and $h:Cto C’$ be morphism of complexes such that $(f,g,h)$ is a morphism of distinguished triangles.

I wonder whether the following statement is true:

If $f:Ato A’$ is a quasi-isomorphism, then there is a quasi-isomorphism $operatorname{cone}(g)stackrel{sim}{to} operatorname{cone}(h)$.

Clearly, if all the involved complexes are concentrated in degree $0$, the statement follows from the snake lemma. Is this true in general?

Many thanks !