I can't prove that the QE (quantifier elimination) is equivalent to this statement:
If $M$, $N$ are two models of $T$ and $N$ is $|M|^+$-saturated, every embedding from a substructure $A \subseteq M$ to $N$ can be extended to an embedding from $B$ to $N$ where $B$ properly extends $A$.
For one side I think since we have QE, every embedding is elementary, so when the new points are added we can take its type over the elements of previous step and since $N$ is saturated the projection of this type will be realised in $N$ so we can extend the embedding. (I'm not sure if this is true...)
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The result you're looking for is Proposition 4.3.28 in Marker's Model Theory: An Introduction.