The concept of phase space was probably floating around for ages, but it only really became central to modern physics in the mid 19th century, with Boltzmann and Maxwell's formulation of statistical mechanics, Hamilton's reformulation of mechanics, and Liouville's theorem.
The concept of phase space is already implicit in the work immediately following Newton, when Maupertuis and Euler studied variational principles. The trajectory in a variational formulation is never isolated--- the neighboring trajectories are important for determining that the solution is extremal. So Euler and Maupertuis were formulating the condition of Newton's laws as a constraint on the space of all trajectories.
The space of all trajectories is the same as phase space, once you impose the equations of motion. But the space itself is not used in these formulations, because it is not explicitly given a full geometrical symplectic structure. All the structure of phase space left implicit in the 18th century work became explicit in the 19th century.
In order to describe a gas, Boltzmann introduced a function $\rho(x,v)$ which described the probability density for a particle to be at position x and have velocity v. This function obeys a dissipative equation, known as the Boltzmann equation, and this is a paradox, as noted by Lorschmidt and others. The paradox is that the irreversibility cannot emerge from time-reversible laws, because you could reverse all the molecular velocities, and make things go back to how they started. So why is Boltzmann's equation irreversible?
The reason was understood quickly--- the equation is assuming that there are no correlations between molecules that enter collisions, so that each one is described by an independent pick from the same distribution $\rho$. This is clearly not right for dense materials, since the collision will imprint the first particle's position and velocity on the second, leading to at least two-body correlation, which might come back to affect the original particle after a few more collisions. In order to incorporate correlations, you have to think of the density $\rho$ as a simultaneous function of all the particle's positions and velocities.
Hamilton's formulation of mechanics shows that it is the momenta that are the correct variables, not the velocities, and this was made clearer with the result that the phase-space volume is conserved in collisions if you use momenta and not velocities as your coordinates. The Liouville theorem of conservation of phase-space volume was a central result in making plausible the derivation of thermodynamics from mechanics, which was most active at the end of the 19th century, but wasn't fully accepted until the 1920s.
Poincare analyzed phase space motions in his famous analysis of the three-body problem and related issues of non-integrability. I do not know this history of engineering, but I assume that by the turn of the 20th century, when Poincare was writing, these results were standard lore.