I'm not at home and so I can't consult my games to check the rules, but here's my first instinct on this.
As you said, at the start of your turn, you are not at 0 hit points, so you won't use a healing surge.
But you will take the 1 hit point of damage during the beginning of your hero phase and drop to 0 hit points.
I don't believe that your turn ends immediately when your hero drops to 0 hit points, so you would continue as normal. So you'd get to save against the poison, you'd draw the encounter card, activate any monsters, etc.
The thing to remember is that the other heroes may have a way to heal you before your turn starts again. In which case, you wouldn't have to use a healing surge.
I'm not 100% on this, of course, as it has been months since I played last, but I'm pretty sure I'm at least close to the correct ruling.