If my bank mistakenly injects $1,000,000 into my account, am I legally allowed to spend it before they realize their mistake?

You never owned that money so legally you are not allowed to spend it..You should contact the bank the moment you realize their mistake..

legally : it  is your money you can spend it with no obligation

but actually : its not your money

Inform the bank immediately and refund amount

I fully agree with the answer been added by MR  badar khan ........................Thanks.

The first professional step is for you to contact your Bank, confirm the source of the fund as you can never tell it might be yours, after that you should know if the fund is yours or not which invariably will aid your next steps. If the fund is not yours, the next step is to ask your Bank to investigation this transaction and do the needful.

I  suppose here the rule of " Errors , mistakes , omission and deletion should be rectified as soon as discovered " must be operate because sooner or later the bank will reach to this accounts mistake , so it better for you to be in the safe side and return the sum that credited to your account  inadvertently and follow the above mentioned rule.

Thanks , with my very best wishes.

legally you have no problem .... but morally you should report immediately to the bank.