  We present the pressure-temperature phase diagram of the antiferromagnet
CeAg$_2$Si$_2$ established via resistivity and calorimetry measurements under
quasi-hydrostatic conditions up to 22.5~GPa. With increasing pressure, the
N\'eel temperature [$T_{\mathrm{N}}(p=0)=8.6$~K] slowly increases up to
$T_{\mathrm{N}}=13.4$~K at 9.4~GPa and then vanishes abruptly at the magnetic
critical pressure $p_{\mathrm{c}}\sim13$~GPa. For the first time, heavy fermion
superconductivity is observed in CeAg$_2$Si$_2$ . Superconductivity emerges at
$\sim 11$~GPa and persists over roughly 10~GPa. Partial- and bulk-transition
temperatures are highest at $p=16$~GPa, with a maximal $T_{\mathrm{c}^{\rm
bulk}}=1.25$~K. In the pressure region of superconductivity, Kondo and
crystal-field splitting energies become comparable and resistivity exhibits
clear signatures of a Ce-ion valence crossover. The crossover line is located
at a rapid collapse in resistivity as function of pressure and extrapolates to
a valence transition critical endpoint at critical pressure and temperature of
$p_{\rm cr}\sim 17$~GPa and $T_{\rm cr}\sim-13~K$, respectively. Both critical
spin and valence fluctuations may build up superconductivity in CeAg$_2$Si$_2$.
