I was having this problem with a new iPhone 6 on iOS 8.0.2. I believe this was working fine before I upgraded to 8.0.2, but can't guarantee that. The problem was with both a VW Jetta Sportwagon and a Fiat Abarth. I also had issues with my Bose headset. On the cars, I got the same behavior. The phone would pair and I could voice dial, but it would run once and drop the bluetooth connection. The call was still connected and I could switch to the iPhone and talked to the person, but not through the car's bluetooth system. On the Bose headset, I got partial functionality though it was buggy and I couldn't hang up a call.
I went to the Apple store. They, of course, were of no help. The guy at the genius (??) bar said he'd never heard of the problem. I suggested he Google it. He then insinuated that Google purposely was making it look like this was a big problem. Oy veh! I then said, the problem could be seen on Apple's own discussion forums. All he had me do was remove some old iPhones and iPads from my Find Your Phone function in iCloud. This did nothing. So, I decided I'd try the one last thing I'd read about -- resetting the phone. I did and it worked! I also noticed that the phone paired differently from the times when it didn't work. This time Verizon showed up as the carrier and before that field would be blank during pairing. The connection is with the Abarth and I haven't tried the VW or the Bose to see if it's working properly.
I'm also told that 8.1 coming Monday will solve this problem. But if you need it now, try the reset. I've heard from others that it works, but I've also heard it doesn't always stay corrected. Fingers crossed.