I LOVE YOUR QUESTION!
Gets the brain juices flowing!
Ok, so first off you are completely right! There is no reason why you should bleed out your margins trying to play that ignorant game. Large companies can afford it because for them is all about volume right, but at the same time you know that not everyone values price above all else otherwise you wouldn't be asking this.

First off - learn about Blue Ocean (I find myself saying that a lot lately) Blue Ocean is not common because is a high level methodology that most so called "growth/business experts" have probably never heard of and much less most companies. You would be surprised. Anyway, learn about it...you might grasp it right away but it will motivate you realizing that it is very very possible.
Second customer service alone won't cut it, but it is definitely a start. My suggestion is look for areas that are lacking involvement, could be a market or industry that most think worthless, remove feature services of your base service list - cutting you costs but not necessarily something that a client would notice and want a discount on. Think hard... this is how you achieve blue ocean.
Blue Ocean is the idea that because is so darn easy to compete on price for commodity products that everyone does it and it becomes traditionally a zero sum game with everyone bleeding each other out in the ocean of an industry and market... where blue ocean comes only after you have figured out what others havent and expand the existing market size by adding previously non-customers to now potential customer and you essentially owning that extra carved market share... you move to the clear blue ocean... sounds like a fairy tale but is very much in practice by many.. I recently did that with a cleaning company.

I can help you figure this out, but it is a long process.
Best of luck and if at least..keep me posted on what you do :)