There is a huge difference between "fulfilling desires" and "happiness". Fulfilling desires is like scratching an "itch". It may feel like we need to. It may feel good while we do it, but scratching just makes the itch worse - and it brings no happiness whatsoever.
Only being of service to others brings happiness. This has been scientifically proven. The happiest people are those who don't live for themselves. And when we say "no" to our desires, we are actually being of service. As Chazal say, "If someone sits and doesn't do an aveirah, it is as if they did a Mitzva". By saying "no", we are being of service to ourselves - to our neshamos, to our (future) spouses, to our (future) kids, and most of all, to Hashem.