Generalization: Euler characteristic with coefficients in a rank $n$ local system equals $n$ times ordinary Euler characteristic. Then using Leray (as in Gunnar Magnusson's comment) gives the case of a finite covering.

The following `sheafy' proof of the generalization should work. Pick an acyclic cover that trivializes the local system and use the associated Cech resolution. An essentially equivalent way would be to do an induction argument using the Mayer-Vietrois distinguished triangle.

Generalization: Euler characteristic with coefficients in a rank $n$ local system equals $n$ times ordinary Euler characteristic. Then using Leray (as in Gunnar Magnusson's comment) gives the case of a finite covering (and more generally that of a fibre bundle).

The following `sheafy' proof of the generalization should work. Pick an acyclic cover that trivializes the local system and use the associated Cech resolution. An essentially equivalent way would be to do an induction argument using the Mayer-Vietoris distinguished triangle.