In the setting of cdgas, this is not true: that is, there is a simple example of a finitely presented coconnective cdga over the rational numbers whose homotopy groups are not finitely generated as an algebra.

Namely, consider the cdga $R$ of (derived) global sections of the structure sheaf ("functions") on the punctured affine plane. Then $\pi_0 R = \mathbb{Q}[x,y]$ and $\pi_{-1} R = \mathbb{Q}[x,y]/(x^\infty, y^\infty)$ where the latter refers to the cokernel of the map $\mathbb{Q}[x^{\pm}, y] \oplus \mathbb{Q}[x, y^{\pm}] \to \mathbb{Q}[x^{\pm}, y^{\pm}]$. (This is easy to check from choosing the standard cover of the punctured affine plane by the complements of the $x$ and $y$ axes, respectively.)

Clearly, the homotopy groups of $R$ are not finitely generated as an algebra, but I claim that $R$ is finitely presented as a cdga. There is in fact an explicit finite presentation of $R$ due to Bhatt and Halpern-Leinster. Namely, let $M$ be the $\mathbb{Q}[x,y]$-module given by $\mathbb{Q}[x,y]/(x,y)$ and consider the natural map $\mathbb{Q}[x,y] \to M$ and its dual $\phi: DM \to \mathbb{Q}[x,y]$, where $D$ denotes Spanier-Whitehead duality in the (derived) category of $\mathbb{Q}[x,y]$-modules. Then the presentation of $R$ is that it is the homotopy pushout of the symmetric algebra of $M$, mapping in two different ways to $\mathbb{Q}[x,y]$, once via the map extending $\phi$ and once via the map extending zero. (As a $\mathbb{Q}[x,y]$-algebra, $R$ has the following universal property: to give a map from $R$ to some other $\mathbb{Q}[x,y]$-algebra $R'$ amounts to the *condition* that $R'/(x,y)$ should be contractible.)