That's a tricky question - but I'll give it my best shot!
I personally think that people who display passive aggressive traits aren't always afraid of conflict, they're just trying to create conflict on their terms.
Rather than approach you directly about an issue they're unhappy about, they'll try to indirectly provoke you into losing your rag first.
My advice for dealing with this situations would be to:
1) Make things as clear and straight-forward as possible. It may sound tedious, but I'd recommend documenting conversations and laying out what you expect in a way that's polite but unmistakable.
Passive agressive people tend to manipulate situations, twist words, and use tools like sarcasm and ambiguity. They lead people on.
The best way to avoid all that is to keep your interactions formal and to the point.
2) Empathise - It may sound like the hardest thing to do, but when confronted with passive aggressive behaviour, the best thing you can do is seem calm and sympathetic.
I used to deal with a lot of angry customers in a previous job, and encountered plenty of passive aggression.
Customers would actively try to provoke me into getting angry, so they could complain about me, write to head office, and receive freebies!
I found the best way to do deal with these situations was to keep calm and remain neutral about the situation. I tried to sound genuinely concerned about their complaints (even when they had no basis), and made it appear like I was doing everything in my power to rectify the situation.
3. Don't take it personally. Most of the time, passive aggressive behaviour isn't your fault, it's something that the person in question needs to deal with themselves.
When you get caught in their trajectory, try to remember that it's not a personal attack, it's just the way they've developed to behave.
That doesn't mean you have to put up with it, but deal with it pragmatically rather than emotionally.