In some text books I've encountered the calculation of electrical work done on a point charge moving in an electric field (e.g. when demonstrating that electric force is conservative) is done by making use of Coulomb's law. However, Coulomb's law is not valid in case of moving charges, and therefore, cannot be used for this purpose. There's clearly something I am missing, but can't figure out what.
Technically you are correct, but the magnetic forces generated by a slowly moving electron (slow compared to the speed of light) are quite small in comparison a robust coulomb force so its a good approximation to neglect the magnetic forces. A good example is the quantum treatment of the hydrogen atom. If you solve the Schrodinger equation for the Coulomb potential, you get very close to the Balmer spectrum. The magnetic effects are called fine and hyper-fine structure because of their small magnitude.
The relativistically correct way of treating a moving electric point charge is by using the Liénard–Wiechert potentials. These incorporate terms with retarded time which is connected with electromagnetic radiation.