If you start brewing because it's a hobby you'd like to get into, then no, you will not save money. Hobbies are activities we enjoy doing and enjoy being a part of... for many reasons, but a lot of the time for no other reason than "it's fun". Not too many hobbies out there that will actually help you save money or make money. (unless you're a rare coin collector and re-seller or something)
Brewing because you want to save money, however, can absolutely be done. Since it's not a hobby and you're not in it just for the fun of it, you don't need, nor want, all the expensive toys. You can absolutely save a ton of money while consuming the same amount of beer. Just depends on how crappy your ingredients are (Madagascar bourbon vanilla bean or Walmart's True Value imitation vanilla) and how ghetto your setup it (stainless steel conical fermenter or random 2 liter Coca-Cola bottles).
Think of it like this. Often times people see a sale and immediately think they're going to save money. "Buy 12 bottles of Pepsi, get 4 bottles free!!!". I only really wanted to buy one $3 bottle, which is more than enough for me. But... if I spend $36, I get 4 bottles for free!
Did I save money? Some will say yes, some will say no. Sure, if I were planning on buying 12 bottles to begin with, perfect. Or maybe in the long run, if I can manage to drink those 12 before they go bad. And buying those 12 will just entice me to drink more than what I was planning on drinking in the first place! But in my eyes, spending $33 more than what I was planning on spending is not saving money, REGARDLESS of what I get in return.
So, if you have that $6 beer once in a while, it may not be worth it to you to spend a couple hundred on a setup, plus $30 on ingredients for 5 gallons if you weren't really planning on drinking that amount anyway...