For abelian varieties this is due to Igusa, I believe (but I have not verified this reference). The argument goes like this, IIRC.

Suppose first that $C$ is a curve over an algebraically closed field $k$. Then for every $d>0$ there are canonical identifications $$H^0(Pic^dC,\Omega^1)=H^0(C^{(d)},\Omega^1)=H^0(C^d,\Omega^1)=H^0(C,\Omega^1)$$ where $C^d$ is he Cartesian product and $C^{(d)}$ the symmetric product. (This is proved in Milne's article on Jacobians in Cornell-Silverman.) Take $d=g$, so that $C^g\to C^{(g)}$ is separable and $C^{(g)}\to Pic^gC$ is birational. Take a $1$-form $\omega''$ on $Pic^gC$ and pull it back to $\omega'$ on $C^g$; then $\omega'=\sum_1^g pr_i^*\omega$ where $\omega$ is a $1$-form on $C$ and $\omega''=\omega'=\omega$ under the identifications in question. Now $d\omega=0$ for reasons of dimension, so $d\omega''=0$ and we are done for Jacobians.

Now suppose that $A=G$ is any abelian variety and take a generic curve $C$ on $A$ (a complete intersection of very ample divisors). Identify $Pic^1C$ with $Alb\ C$. Then the natural map $Alb\ C\to A$ induces an injection on global $1$-forms ($C$ is in general position on $A$) and so on $2$-forms. Then for a $1$-form on $A$ we have $d\omega=0$ on $Alb\ C$ and then $d\omega=0$ on $A$.

For abelian varieties this is due to Igusa, I believe (but I have not verified this reference). The argument goes like this, IIRC.

Suppose first that $C$ is a curve over an algebraically closed field $k$. Then for every $d>0$ there are canonical identifications $$H^0(Pic^dC,\Omega^1)=H^0(C^{(d)},\Omega^1)=H^0(C^d,\Omega^1)^{{\frak S}_d}=H^0(C,\Omega^1)$$ where $C^d$ is the Cartesian product, $C^{(d)}$ the symmetric product and the superscript ${\frak S}_d$ indicates the subspace of invariants under the symmetric group ${\frak S}_d$. (This is proved in Milne's article on Jacobians in Cornell-Silverman.) Take $d=g$, so that $C^g\to C^{(g)}$ is separable and $C^{(g)}\to Pic^gC$ is birational. Take a $1$-form $\omega''$ on $Pic^gC$ and pull it back to $\omega'$ on $C^g$; then $\omega'=\sum_1^g pr_i^*\omega$ where $\omega$ is a $1$-form on $C$ and $\omega''=\omega'=\omega$ under the identifications in question. Now $d\omega=0$ for reasons of dimension, so $d\omega''=0$ and we are done for Jacobians.

Now suppose that $A=G$ is any abelian variety and take a generic curve $C$ on $A$ (a complete intersection of very ample divisors). Identify $Pic^1C$ with $Alb\ C$. Then the natural map $Alb\ C\to A$ induces an injection on global $1$-forms ($C$ is in general position on $A$) and so on $2$-forms. Then for a $1$-form on $A$ we have $d\omega=0$ on $Alb\ C$ and then $d\omega=0$ on $A$.