When you're considering the properties of gases there are often two ways to look at the problem. The first is to use the continuum approximation leading to the usual laws like Boyle's law, Charles' law etc. The second is to treat the gas as many tiny particles (i.e. the gas atoms/molecules) and use Newtonian mechanics. In this case I think the second way is to understand what's going on.
The rocket motor burns a mixture of fuel and oxygen to produce a very hot gas. By very hot we mean that the gas molecules have very high random velocities:
This diagram is supposed to show a representative sample of the atom/molecules in the flame. They are all moving in random directions, so the total momentum of all the atoms is going to be close to zero. This means burning the fuel has not changed its momentum - this may seem a funny thing to say, but bear with me.
If the fuel were burning in a vaccum the random directions of the atom velocities would mean the ball of atoms expands in a roughly spherical way and the total momentum stays zero. But the fuel is not burning in a vacuum, it's burning inside a combustion chamber:
The reason this matters is that the atoms can't escape to the right or up or down because the walls of the combution chamber are in the way. So they will bounce around until some random collision (with the walls or other atoms) gives them a velocity pointing to the left:
So very quickly all the atoms are going to end up with their velocities pointing in roughly the same direction, because at that point they can escape from the combustion chamber and go flying off into space. Now let's calculate the momentum of all those atoms. If there are $N$ atoms and the mass of each atom is $m$ and their average velocity is $v$ then the total momentum is now $Nmv$ (we'll take velocity to the left to be positive). The momentum of the fuel before burning was zero, and after burning it's $Nmv$, so the momentum has changed by $Nmv$. Conservation of momentum means the rocket must have changed its momentum by $-Nmv$ so that the total momentum change adds up to zero.
So burning the fuel and allowing it to escape to the left means the rocket must have accelerated to the right. In other words the rocket engine has produced a force on the rocket, and we've calculated this without needing to think of pressures or other macroscopic quantities. In fact we can be more precise about the force. If the rocket produces $N_s$ particles of exhaust gas per second then the momentum change of the rocket per second is $-N_s mv$. Momentum change is force times time, so the force on the rocket is simply:
$$ F = N_s mv $$
This force is produced simply because atoms moving to the right bounce off the end of the combustion chamber, and hence push the rocket to the right, but atoms moving to the left don't.