During the last CM session I failed to recall immediately how the volume conservation can be derived from Hamiltonian equations. There is a beautiful explanation in Tong's notes on dynamics and relativity - the credit goes him.

To spare you reading through the whole notes (if you have time, I really recommend it ;)), consider the evolution of a small volume in dimensional phase space:

The coordinates will evolve in time according to:

meaning that:

It then follows from the substitution rule for multivariable integral that:

where is the Jacobian determinant:

to get the last equality, observe the fact that only the diagonal does not pick up any dependence and the terms vanish by antisymmetry. Therefore:

and the phase-space volume remains unchanged in time.