That's a special case of Noether's theorem for space-time translations, which is a symmetry of Minkowski space. The corresponding conserved quantities are energy and momentum. For fields it defines the canonical energy-momentum tensor
$$\Theta^{\mu \nu}=\frac{\partial \mathcal{L}}{\partial (\partial_{\nu} \phi)}\partial^{\mu} \phi-\mathcal{L} g^{\mu \nu}.$$
The momentum density components are given by ##\Theta^{0j}## (##j \in \{1,2,3 \}##). Now it should be easy to show the above formula.