The simplest physics description of potential barrier tunneling can be found here:
According to classical physics, a particle of energy E less than the height U0 of a barrier could not penetrate - the region inside the barrier is classically forbidden. But the wavefunction associated with a free particle must be continuous at the barrier and will show an exponential decay inside the barrier. The wavefunction must also be continuous on the far side of the barrier, so there is a finite probability that the particle will tunnel through the barrier.
Note it is the same energy level occupied by the particle inside or outside the barrier, it is the wavefunction $Ψ$ that is affected, i.e.the probability $Ψ^*Ψ$ of finding the particle at a particular x, in this one dimensional representation.
It is all quantum mechanics after all, a matter of probabilities, there is no change in energy so there is no problem.
Edit after comment:
In the image above, the energy is negative counted from the top of the well and continues negative counted from the top of the well,or positive, if the zero is defined at the bottom of the well.
, I am under the impression that having a particle inside a potential barrier would make for a higher energy state
A change of energy states can happen only with the exchange of energy, so it will no longer be tunneling, but another quantum mechanical proccess.