Given that matter cannot escape a black hole, how did the big bang produce the universe we see today? Extrapolation of the expansion of the Universe backwards in time using general relativity yields an infinite density and temperature at a finite time in the past.
If the matter contained within our galaxy were concentrated within a small radius wouldn't that lead to the entire universe being a black hole? If so how is it that all the matter of our observable universe could have originated from a region of infinite density?
 A: During the initial stage of the Big Bang known as the inflation stage (hence the name "Inflation Theory") the laws of physics were different than they are today. The foces of electromagnetism, the strong and weak nuclear forces and gravitation were unified in a single force.
In the first 10-35 seconds the forces separated and the universe as small as it was at this stage, filled with what's known as the vacuum energy. As a consequence of this vacuum energy density (which plays the role of an effective cosmological constant), gravitation became repulsive for a period of about 10-32 seconds.
This caused the universe to expand an an astonishing rate.
By the time gravitation phased towards becoming an attractive force, the universe has expanded beyond the critical density required for it to collapse back in on itself to form a singularity.
A: 
yields an infinite density and
  temperature at a finite time in the
  past.

Not quite, the extrapolation stops sometime around a Planck length, nobody is saying what happened before then

wouldn't that lead to the entire universe being a black hole?

No, even if the universe was originally a singularity after inflation it wasn't and isn't
A: Rather belatedly, let me mention my answer to Did the Big Bang happen at a point?. This explains the geometry of the expanding universe in laymans terms, and should make it clear how it differs from the geometry of a black hole.
because the geometry of the two solutions is completely different there is no reason to expect them to evolve in the same way, and indeed they don't.
A: This is actually a common question.  Many websites have been setup to try to explain this.  I like this one for instance.  I shall attempt to do my own layman explanation.
First of all, in order to have a black hole, you need to have a place for it to be in.  Since there was no such thing as a universe, there isn't a place for the black hole to actually exist in.  It's like asking, what is north of the north pole.  There is no reference for an answer.
Secondly, gravity and other fundamental forces didn't act the way we are used to them acting.  All four fundamental forces were combined in one basic force.  Therefore there was no such thing as gravity to actually act on the mass as it existed.  And then there is the problem that there was no actual mass.  It was energy, which was creating the density of the universe.  I know it's kind of counter-intuitive to how we are used to understanding these terms.  Actually, Brian Greene has a good explanation of all this in his book Elegant Universe.  Suffice it to say, at the start of the universe, the fundamental forces really acted very differently from how we see them now.  As gravity was separating itself from the other forces, it actually had a repulsive effect as opposed to attraction.
Finally, the big bang wasn't really an explosion or a bang as one would think of it.  It's actually the rapid expansion of space itself.  Instead of things themselves speeding away from each other in a fixed space, the motion is actually caused by the space between the objects themselves getting bigger.  Visualizing this is usually presented as a balloon expanding, and seeing how two dots on the surface of the balloon get further away from each other without actually moving on the surface.  Of course, it's harder to visualize this in three dimensions (or actually four).  While we are limited to the speed of light for any objects, the expansion of space itself is not limited by this (Brian Greene's book also has info on this).
The bottom line is that anything that you think you know should probably be discarded when thinking about the big bang.  The reason that there is "string theory" and many other things is that classical quantum physics and relativity physics break down at the Plank Epoch which is at the heart of the big bang.
I hope that helps.
A: Good question! I often pondered that myself. As this website explains, the Big Bang wasn't a black hole basically because it couldn't be! A black hole is the mathematical solution to Einstein's equations of General Relativity that describes a pre-existing region of spacetime that has gravitationally collapsed and formed a singularity. Since there was no spacetime before the Big Bang, it couldn't possibly be a black hole.
But what was the Big Bang? It turns out, if you flip the sign of t (time) in the black hole solution to Einstein's equations, what results is also a perfectly valid mathematical solution. It describes a black hole in reversed time, or in other words, well, the Big Bang! A singularity explodes outwards into a nice, flat(tish!) region of spacetime. This solution is also known as the white hole solution.
A: *

*Inflationary model is only part of the answer.

*since the mass-energy in the universe was evenly distributed at the end of inflation, there was no over-density in one region that resulted in black hole.  This can be explained by Newton's shell theorem.

*separately, radiation exerted an outward pressure that prevented over-dense regions from forming.  Newtonian dynamics goes a long way towards explaining how the outward pressure of radiation balanced the collapsing force of gravity.


Analogy:


*

*temperature of a newly formed neutron star is 10^11 Kelvin

*temperature of universe after inflation was 10^27 Kelvin.



Inflation theory was proposed to resolve the flatness problem and the horizon problem.  Inflation ends after the first 10^-32 seconds.  Our observable universe was the size of a grain of sand that contained a mass-energy ~10^80 GeV, which is much denser than the critical density that would presumably make it collapse into a black hole.
However, it seems to me that if the observable universe was all there was (just this grain of sand surrounded by vaccuum), then the universe would have collapsed to a black hole.  The universe must have been much larger than our observable grain of sand, and mass-energy must have been smoothly distributed outside of our observable grain of sand.
Here is one good website for understanding the dynamics.  I recommend starting with Newtonian dynamics section before delving into GR and FRW dynamics.
 http://hyperphysics.phy-astr.gsu.edu/hbase/Astro/expuni3.html#c1

Some insight into the expansion can be obtained from a Newtonian
  expansion model which leads to a simplified version of the Friedman
  equation.

A: There is a concept known as entelechia, which basically means an idea so complicated and so full of border cases that no useful ideas can be obtained from it.
One entelechia is the concepto of god, a súper being who created everything in just 6 days and who needs your money. And the fossil record was put there by good just to test your faith. Eventually it becomes so complicated that it is extremelly unlikely that there is a god.
The same occurs with the big bang theory. If there was ever a big bang, it means that at some point in time a place where all matter in the universe was in a place smaller than any star. And cosmology shows that any star with the size of the sun and 1,000 times its mass is going to be a black hole. And a black hole by definicion will not allow anything to escape, not even light.
So you see, you can complicated the theiry as much as you want, change theblaws of physics when the big bang ocurred, create inflation theories, but how is that any different than saying that the fossil record was put in there to test your faith?
