It is possible to obtain the canonical basis from the PBW basis, as long as you are working in the quantum group U<sub>q</sub>(n). The canonical basis seems to be inherently a quantum phenomenon, so it shouldn't be surprising that we want to compute in the quantum group.
If E<sub>π</sub> is an indexing of the quantum PBW basis, then what one first proves is that applying the bar involution is unitriangular with respect to this basis. There is even a mathoverflow question about this: https://mathoverflow.net/questions/110108/.
Now it is a matter of linear algebra to prove that there exists a unique bar invariant basis b<sub>π</sub> such that
$$b_\pi=\sum_\sigma c_{\pi\sigma} E_\sigma$$
with c<sub>σσ</sub>=1, c<sub>σπ</sub>∈qℤ[q] if σ≠π and c<sub>σπ</sub>=0 unless σ≤π. This basis b<sub>π</sub> is the canonical basis.