  For a two-dimensional Artin group $A$ whose associated Coxeter group is
hyperbolic, we prove that the action of $A$ on the hyperbolic space obtained by
coning off certain subcomplexes of its modified Deligne complex is
acylindrical. Moreover, if for each $s\in S$ there is $t\in S$ with $m_{st}<
\infty$, then this action is universal. As a consequence, for $|S|\geq 3$, if
$A$ is irreducible, then it is acylindrically hyperbolic. We also obtain the
Tits alternative for $A$, and we classify the subgroups of $A$ that virtually
split as a direct product. A key ingredient in our approach is a simple
criterion to show the acylindricity of an action on a two-dimensional
$\mathrm{CAT}(-1)$ complex.
