It's possible that this can be achieved by doing things differently, but here's what I found worked.
Let's assume that the airlock you start next to (the southern most one) is number one and you follow the dome around to the right (counter-clockwise) and count to six. I opened the doors in the following order:
1 then 4
2 then 5
3 then 6
Edit: Xx UponDoom Xx made a great guide with a picture of what I'm talking about, for those confused. Some have mentioned that the order doesn't matter, while it did for others. All I can tell you is that it DID matter for me!