(Using the notation from Chang-Keisler.) Let the models be $M_1 , \ldots , M_n \subseteq \mathscr{S}$.
For each sentence symbol $S$ of $\mathscr{S}$ and each $i \leq n$, by $S_i$ denote $ \begin{cases} S, &\text{if }S \in M_i \\ \neg S, &\text{if }S \notin M_i. \end{cases} $ Let $\Gamma$ be the set of all sentences of the form $ (S^1_1 \wedge \cdots \wedge S^m_1 ) \vee \cdots \vee (S^1_n \wedge \cdots \wedge S^m_n ) $ where $S^1 , \ldots , S^m \in \mathscr{S}$.
Clearly each $M_i$ is a model of $\Gamma$.
Suppose now that $N \subseteq \mathscr{S}$ is some model distinct from each $M_i$. For each $i \leq n$ there is a sentence symbol, $S^i$ such that either $S^i \in M_i$ and $S^i \notin N$, or the opposite. It is easily seen that $N$ is not a model of the sentence $ (S^1_1 \wedge \cdots \wedge S^n_1 ) \vee \cdots \vee (S^1_n \wedge \cdots \wedge S^n_n ) $ ($N$ cannot model $(S^1_i \wedge \cdots \wedge S^n_i )$ since by construction it does not model $S^i_i$.)