Is energy always conserved?

I'm not sure about conservation of energy, but in the case of cosmological redshift, I can't figure it out. Doesn't the energy, for lack of a better word, "expand" as well?
For the sake of simplicity, let's not talk about light from a galaxy, let's think about a simple short laser beam. Like a ten-minute blue laser beam, and let's pretend it is far enough that it changes color and we can somehow see it over the distance. The redshift occurs because the spacetime is stretching between us, but isn't it equivalent to the source of the laser having a relative speed? In that case, there would be a time dilation and the source's 10 minutes would be longer for us, and we detect like 12 minutes of lesser energy, so the extra time would make up for the red shift. Am I wrong?
Or is the cosmological redshift fundamentally different than Dupler effect?