A small amount of the static picked up is from the big bang (or, at least, from the few hundred thousand years after the big bang when the universe finally became transparent to photons). But the static on your TV isn't really the supporting evidence for the big bang, it's more of an inference about how much of the noise picked up by your TV is leftover based of of better measurements of this radiation made elsewhere.
Now, this radiation really is a very compelling piece of evidence for the big bang. It's called the Cosmic Microwave Background Radiation. The theory makes a very precise prediction about what this "leftover" radiation is supposed to look like. A few years ago, a satellite called COBE was used to study this CMBR and it was found to match the theory to within experimental error (in other words, the theory was dead-on correct). That's the "leftover" radiation which provides the conclusive evidence for the big bang, but your television static wasn't making a good controlled measurement of it of the kind needed to prove the theory. You can read about it here:
http://en.wikipedia.org/wiki/Cosmic_microwave_background_radiation
Some of the best evidence for the big bang is the abundance of certain elements created by Big Bang Nucleosynthesis. When the universe was very young, it was also very dense and hot. The theory predicts that there are certain elements which would have been created by nuclear fusion during this time. The theory also predicts the abundance of these elements. The observations of these elements in the predicted amounts is good evidence for the big bang because 1) the big bang model got them right and 2) no other theory does that.
You can read more about big bang nucleosynthesis here:
http://en.wikipedia.org/wiki/Big_Bang_nucleosynthesis
and here:
http://scienceblogs.com/startswithabang/2009/08/how_sure_are_we_that_the_big_b.php
Regarding a couple of posts downthread:
The CMBR radiates as a blackbody across a large, large range of frequencies. Its peak frequency is well outside of the range of UHF antennas, but this isn't the only frequency found in the CMBR (which has easily detectable radiation in the 0.3 GHz to 300GHz range). UHF ranges from 300MHz to 3GHz, so that it is entirely capable of picking up part of the CMBR.