You were in the RIGHT.
On the other hand, before I personally judge the owner of the dachshund, sometimes little dogs just sneak out the dog. Still, if it ran away, it's probably all right.
My three "big" dogs cocker spaniel, cockapoo, border collie/cocker started picking on my daughter's chihuahua. It was because the chi thought it was a pitbull and would challenge the dogs with a stare. One day my mild mannered border collie/cocker mix started tearing into her and the others went into pack mentality and joined in. By the time I saved the chihuahua, I was bitten up and the chi had one eye pulled almost out of it's socket. My dogs have never even nipped me before or since. My easy child daughter was screaming. In that frame of mind, shaking, easy child and I drove the chi to the vet who reparied the eye with surgery. Since that incident, both my border collie/cocker and my cocker spaniel can not be around smaller dogs. We rehomed the chi.
It never occurred to me though that the dogs were "bad" or "evil." It just struck me as dogs picking on a smaller dog. I don't know if we can ever get the two offenders to be nice to small dogs. We do have a trainer who has helped in other areas, but I'm afraid to try the cocker and his sister with small dogs ever again. Because I must take precautions, I walk each dog seperately so I can maintain control of my dogs if they go nutty over another dog (in a bad way) and my dogs stay in the small dog park at the dog park. And if anyone tries to come in, I warn them to let me get my dogs back in the car first.
I wish I didn't have to do these things, but that's the way it is for my pups. They are sweet pups in the family, but not good with other dogs. BUT...they ARE animals and I love them to death, even though they do not always have good manners with other canines.
I don't know if this helped or not...it was my attempt to offer some empathy. I'll bet that little dachound, which are feisty little things, challenged your dogs with his eyes and they took the bait.