 Solution

A structure $M$ is $\kappa$-saturated when every type over a parameter set of cardinality below $\kappa$ is realized in $M$. It is $\kappa$-homogeneous when every partial elementary map of size below $\kappa$ has the one-point extension property.

Let $f:A\to B$ be such a map and $c\in M$. Transport $\mathrm{tp}(c/A)$ through $f$ to a type over $B$. Its parameter set has size below $\kappa$, so saturation supplies a realization $d$. Then $f\cup \{(c,d)\}$ is partial elementary. This proves that saturation implies homogeneity.