As it happens this issue has just been mentioned in the Science Fiction Stack Exchange. The simple answer is that we cannot know that there is anything outside the observable universe. The best we can say is that it seems likely.
We approximate the universe using the a spacetime geometry called the FLRW metric. This is based on the assumption that the universe is the same everywhere - technically that it is homogeneous and isotropic. If the universe is the same everywhere then obviously it's the same beyond the bits we can see, but remember that this is just an assumption.
It's possible to construct a universe that looks like an FLRW universe locally but contains no matter outside some boundary. This metric is called the Oppenheimer-Snyder metric and was devised as an approximate description of a collapsing star forming a black hole. However we can reverse the time direction and the metric then describes matter emerging from a white hole. As long as the boundary is farther away than the edge of the observable universe we would not be able to tell the difference between an Oppenheimer-Snyder universe and an FLRW universe.
However the Oppenheimer-Snyder construction is a rather artifical one. It is created by welding together a patch of spacetime described by the FLRW metric and a patch described by the black hole (Schwarzschild) spacetime. While it's technically possible for this geometry to describe our universe it would take a deity with a rather peculiar sense of humour to arrange the universe in this way. The FLRW universe seems much less contrived and therefore more likely.
You're right that the surface of last scattering (SoLS) is the farthest we can see in practice. This light is seen as the cosmic microwave background (CMB), observed e.g. with the Planck spacecraft.
The term "observable Universe" refers to the farthest we can see in theory, and is defined as the distance a photon is able to travel in the time from the Big Bang (BB) to now, in the hypothetical event that is does not interact with any other particles. But since the Universe was filled with free electrons — which scatter photons of all wavelengths effectively — until the CMB was emitted 380,000 years after BB, this does not happen in practice.
If at some point we will be able to measure the cosmic neutrino background, which decoupled from matter 1 second after BB (e.g. Fässler et al. 2016) and interacts extremely weakly with matter along its journey, or primordial gravitational waves which are thought to have been emitted during inflation, $10^{-32}$ s after BB, and which doesn't interact with matter at all, this will come from the "edge of the observable Universe"$^\dagger$ (the so-called "particle horizon").
Due to the expansion of the Universe, light from the SoLS is redshifted by a factor of 1100, while (hypothetical) light from the particle horizon is infinitely redshifted. The distance to the SoLS is roughly 45.6 Gly (billion lightyears), while the distance to the particle horizon is slightly larger, 47.1 Gly. The reason that the difference between the two radii is not only 380 kly (the distance that light travels in the 380 kyr that went before the CMB was emitted) is that the expansion rate at that time was much larger (at $t=380\,\mathrm{kyr}$, it was $H = 1.4\times10^6\,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$, compared to today where it is only $H_0 = 67.8\,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$).
$\dagger$Although neutrinos probably have mass and thus don't travel quite at the speed of light.
Best Answer
Science is all about predictive power. It is entirely possible that the laws of physics are completely different from the ones we know. The universe could be managed by tiny deamons and are just waiting for someone to sound a trumpet before the walls come down. However, there's no evidence to suggest we can make predictions in this way.
What we can say is that every observation we have made is consistent with the universe having no center. If we make predictions based on this assumption, we have a curious tendency to be right.
There's nothing that prevents there from being a "center" elsewhere, if the laws of physics still resulted in the same set of observations that we see. We tend to ignore this because the results are more complicated, and they don't provide any better predictions.
To borrow from Russel's Teapot, I can predict that balls fly through the air in a (roughly) parabolic arc. I can also predict that balls fly through the air in a parabolic arc and there is a teapot orbiting around Jupiter. Unless I can make observations around Jupiter, the second theory doesn't add any more predictive capability, so we can side step it entirely.
In the case of the idea that the universe has no center, we can stick to that simplistic notation until someone finds out how to observe something outside of the observable universe. Obviously this phrasing has some drawbacks...