Since when does limited resources mean you can turn down a request with no further explanation? If it was a good enough argument in and of itself, that's what you should be able to do with it. Every department has limitations, but the physical devices make IT different, and yet, that's not enough by itself.
You have to communicate it in a way that is as understandable as necessary (meaning, the more clout you've created for yourself, the more concise it can be). If you create a company purchases policy, whether by your own power, or by drawing on the power of executives, then you can point to that as necessary.
But if you haven't even checked to see if there is such a policy, you're screwed.

Other departments have to juggle similar concerns when it comes to overtime compensations, so get a framework for handling the constraints.
Don't make it your job to defend your position, most likely it really isn't - and defending is a losing game - it puts your defeat into the objective zone!

As for your example: it is the finance director's job to weed out frivolous requests. Every department gets the same treatment there, unless they have a card from the CEO that says "This boy is golden, give him whatever he wants". Other managers just don't go griping about those "stupid questions" because they know it's their job to sell their requests. The finance director knows that, yes, defeating finance director resistance IS one of the objectives of managers. They can live with it. So can you.
The business is best served when the finance director maximally resists requests and managers have to maximally drive their requests through finance resistance - odd, perhaps?

But look at it this way; lets say the company has an IT policy that says "IT gets whatever they need to make everybody happy". Cool, right? But then you don't need an IT manager, just a bunch of happy techs and developers and someone to collate shopping lists.

The need for an IT manager stems from the assumption that a managed IT department is able to enable business well enough, for less. Having to stand up for your requests is a natural part of that equation.

If you don't know exactly why the other group can't pick up your slack, then it must mean that maybe they can. Think about that for a second: it means, that if it is absolutely certain that they absolutely can't, that's what you have to say, and that's what you have to be able to explain. Simple as that.