I agree with everyone else that you should probably get some advice (medical and instructor), however I also wanted to add that for me this got better over time as my core strengthened.
However, I still need to be really careful and mindful, especially in cobra and up dog. Three days ago, I muscled into too high a cobra without really thinking and clearly did some damage. What I find really soothes that area when I do overdo it is lying on my back, hugging my knees into my chest and rocking side to side to massage the area - helps gently release some of the muscles.
Also what I sometimes find can be a factor is my own inner narrative about pain and again, more yoga practice and my meditation practice has really helped with this.
Last night, I was cooking and my narrative was really waxing lyrical about how painful my back was and how stupid I was for not being mindful with the cobra. I caught it, stopped and put a lot of attention in my lower back and asked "is it painful?" and I realised after some exploration that it wasn't really pain, it was a bit achy in that niggly sort of way - discomfort but not pain. I was able to drop the narrative, relax and carry on. By late evening, I had no pain in my back at all, so I think narrative and tension was the main culprit.
However (and this is the big caveat), I could equally have paid attention and discovered yes it was sharp pain and I'd done some serious damage needing medical. Or that it was dull, muscular pain and I needed to take some restorative rest. It's the paying attention that's the important bit :)
That last part sounds a bit woo, but I do find that over time yoga helps you to listen to your body more effectively so you can make better judgement calls about injury and soreness.