Use effective when you mean producing a desired effect.
Use efficient when you mean productive or with little waste.
Yes:
Cooling a desktop computer with liquid helium is effective, but it isn’t efficient.
(Helium cools very well, so it’s effective. However, helium is more expensive than other ways of cooling a computer, so helium isn’t efficient.)
An efficient power supply has a higher effective power output than an inefficient power supply.