We establish a scaling limit result for the length $\operatorname{LIS}(σ_n)$ of the longest increasing subsequence of a permutation $σ_n$ of size $n$ sampled from the Brownian separable permuton $\boldsymbolμ_p$ of parameter $p\in(0,1)$, which is the universal limit of pattern-avoiding permutations. Specifically, we prove that \[\frac{\operatorname{LIS}(σ_n)}{n^α}\;\underset{n\to\infty}{\overset{\mathrm{a.s.}}{\longrightarrow}}\; X,\] where $α=α(p)$ is the unique solution in the interval $(1/2,1)$ to the equation \[\frac{1}{4^{\frac{1}{2α}}\sqrtπ}\,\frac{Γ\big(\tfrac{1}{2}-\tfrac{1}{2α}\big)}{Γ\big(1-\tfrac{1}{2α}\big)}=\frac{p}{p-1},\] and $X=X(p)$ is a non-deterministic and a.s. positive and finite random variable, which is a measurable function of the Brownian separable permuton. Notably, the exponent $α(p)$ is an increasing continuous function of $p$ with $α(0^+)=1/2$, $α(1^-)=1$ and $α(1/2)\approx0.815226$, which corresponds to the permuton limit of uniform separable permutations. We prove analogous results for the size of the largest clique of a graph sampled from the Brownian cographon of parameter $p\in(0,1)$.