Who has the responsability
Who has the responsibility to start and finish the Unit of work in a MVC architecture?
As zihotki said you would be violating the SRP if you give this responsibility to the controller. This is a data manipulation oriented pattern, and as such should not be a concern for the controller ... that would make it two violations: one for the SRP and anothrt for the SoC principle.
As for who has the responsibility, that's something to be defined by your architecture. The StartRequest/EndRequest suggestion seems solid enough.