Moose are easy to kill, but they are tough to get to fall over and stay "falled over".  Cartridge is not the key here either.  The key is hitting them right.  I'll probably get flamed, but I shoot 'em through the shoulders with a 300 WM and 168 gr TSX bullets.  Busted shoulders puts them out of commission real quick like, and if they "need" another one for the sake of humanity, go for it.  I hate chasing a wounded moose, and I also hate to have to deal with one even a hundred yards into a swamp.  I prefer to deal with them in a cutover, kinda like Cat's in the pic above.
And they don't always give you the classic broadside shot either, so decide where the most vitals are and aim for their.  You may need more than one round.  Someone else said that often the first shot is lethal, just that moose are too dumb to lay down -- I'd say not to DUMB -- just too BIG.  A mature bull or cow can easily go 1,200 pounds.  Most elk don't get that big, I think.
Anyway, like I said initially, take a good 160 ish bullet - a tad smaller if it is a TSX, and hit the back woods.
Happy hunting!