I've done this very successfully with my extended group of players. I've played a 1e Tomb of Horrors run and a two year BECMI campaign.
The first thing to do is pitch it to them and be enthusiastic! "This is a classic scenario and we should play it the way it came about!" Dark Sun is one of the most unique D&D settings I've played in, and it was envisioned in 2e. If you want to play the original, the way people originally thought about it, you should play 2e just so you understand where it came from.
Second, acknowledge things are going to be different. 2e is different from 1e is different from 3.xe is different from 4e. Things are going to change and people will have the most fun by embracing that change. Even if people start out campy, that's ok. One of the most well-developed characters in my BECMI game was a halfling with nothing mechanically different from any other halfling, but it worked because the player started out by accepting the mechanical limitations and working from there.
Third, stand firm. If you want to run 2e Dark Sun, stick to your guns rather than convert. I told folks I was really interested in running ToH in 1e to be authentic, but I wasn't going to convert it. If they wanted to experience it (and here's why they should WANT to) that was the way it was going to happen.
Last, listen to real concerns. A core element of any edition is house-ruling the bits your players are most concerned with. If your players, after having tried something out, have problems with some element of the system: change it. In BECMI, we went through a few different versions of a combat round to get with something that really felt right. Amusingly, it was closer to the original group initiative than anything else, but what was important was that all the players and I sat down and talked about what we liked and didn't like and made changes we agreed on. This should happen in any edition, but it seems easier in older editions without as much focus on Rules As Written.