I am not sure whether "put on" is the right one as I am getting anything inside a DVD player or any other device. Nor I do believe the verb play is suitable in this particular case.
Here are two examples.
Hey dad! Let's play Monk on Netflyx
Hey meemaw! Let's put on Monk on Amason Brime.
And what if it was something I have to do before doing something else?
Here is the example: I have to put on/play Monk on my mother's television for her to watch it before I go to work.