Xを超越している usually means something cannot be measured/analyzed by the existing framework of X. 次元を超越している means something cannot be analyzed in terms of "dimension", which, as you say, generally refers to time and space ruled by the physics of our world. Therefore, we can say this world (天界) is not governed by the ordinary rules of physics, but the phrase by itself does not tell if the world is a spiritual world or just a parallel world. It does not necessarily say the world is "higher" than our world, either.
Conceptually, 次元を超越している is different from 高次元 (high-dimension) such as the 1024-dimensional space. When you say 高次元, you are discussing something within the framework of 次元. 次元を超越 refuses such an analysis