What part of our brain controls our involuntary actions?

It would be better to ask what part(s) control voluntary action. In general, everything is involved with involuntary actions.

If you mean muscle activity (actions), then the typical answer is the involuntary nervous system, specifically the autonomic nervous system, in which there are two main parts (as determined by nervous transmitters and function). (For example, the Sympathetic nervous system.) But there are chemical responses controled too.