There's no evidence, it's theory. Physicists believe that Dark Matter is what makes the universe expand. They can't observe it, they can't test it, they can't hold it, they just have to believe that it exists because if it doesn't exist, a lot of other science has to be thrown out. It's a consequence of a lot of other things.
Dark Matter is a question of faith for physicists. If you don't believe, then you're bucking the system because without dark matter, a lot of the science of the last 50 years starts to fall apart.
To me, that's no different than believing in God, except I think a belief in God has more positive outcomes.
You've confused "dark matter" and "dark energy". Dark matter is used to explain why galaxies rotate, rather then fly apart given their mass and angular rotation. Dark energy is involved in the universe expanding.
As you indicate, dark matter and dark energy are just constructs to make the models match reality. It's not a question of faith. Any scientist would love to know what it really is. Anyone who comes up with a good theory to either explain dark matter or energy, or can show why they aren't needed, are a shoo-in for a Nobel prize and aren't bucking the system. Such a theory would be examined carefully and very skeptically, due to the implications on many other parts of science.
In some ways, science is in a similar position as a little over a century ago. There were observations that couldn't be explained by classical mechanics, but needed "new" ideas such as quantum mechanics (explained some features of the photoelectric effect and black body radiation) or relativity (orbit of Mercury, the Lorentz-FitzGerald contraction). Dark energy and matter are the same sort of thing- placeholders until we get a better idea.
It's not like believing in God since we know dark matter/energy is nothing more than a construct that we use until we know more about how God made the universe (or we know more about how the Universe came into being if one chooses not to involve a deity).