The A350 & A380 (and all current Airbus airliners) use bleed air from the engines, just like all other airliners bar the 787.
The reason that people feel better after long flights on more modern aircraft like the A380, A350 and 787 than they do on older aircraft is due to lower cabin altitude and thus higher cabin humidity, which is largely down to the use of more composites in the construction, which can handle the higher humidity without corrosion problems. This better cabin environment is possible with bleed air or with electric compressors.
As an aside, the 787 also offers some protection from "fume events" (leakage of combustion byproducts or oil fumes from the engine into the cabin) since it doesn't use bleed air. However, there's still oil in the electric compressors so there's still some (much diminished) risk of fume events.