Like most corporations, finances are split up by operating departments. At the department level, everyone has a budget and a balance sheet. So the University as a whole has an operating budget, which is a combination of tuition/fees, income generated by the general endowment, and scholarships, grants, revenue from the hospital, etc. In many cases, there are specific sources of funds allocated to specific expenses. Endowed chairs, living scholars, naming donations for schools, etc. A person could give a piece of land and stipulate that it not be sold for 50 years.
Basically, there is a general pool of funds upon which to draw income, and then funding (mostly donations) with stipulations. A donor could endow the head coaching position, in theory. The donor pays $X and the HC becomes the "Rich Guy That Gave a Bunch of Money Head Coach at The U." The Living Scholars program is a good example of a stipulated donation.
Back to operations. The University has a budget, then the Athletic Dept, and then each sport. At basically every school that plays sports, the only sport that makes money is football. I'd guess Duke makes money off basketball, if anyone. As stated by other posters, there are Title IX stipulations dictating fair allocation of resources, which is an encumbrance on the football budget. Point being, the University allocates money to the Athletic Dept as a pool of funds, and will have some say as to how said funds are allocated to individual sports. As far as sponsorship, I don't know how Title IX impacts how those funds are allocated. Regardless, it stands to reason an argument can be made to allocate a greater percentage of sponsorship money to football, as it attracts by far the greatest audience, which is what the sponsor pays for.
When discussing "the University opening up its purse strings," there may be some impact as to the general allocation of funds to the Athletic Dept, as well as specifically to football, within the parameters of Title IX. But what we are likely seeing is the confluence (**** you Al Golden) of 1) a WAY better AD than we've had in a long time, 2) some increase in University funds to football, 3) increased donations, 4) the Adidas deal, 5) increased ACC revenue and 6) other factors. Basically, what we are myopically referring to as "The Administration."