I'm working in 2.79, so hope it's the same in whatever version you have.
If you've done a simulation you are happy with, you can "Bake from Cache". Then use the "Make Vertex Parent" function:
Select the object you want to have follow the simulation, then select the cloth object. Then go to Edit mode, and select the vertex nearest the object that is to follow the simulation. Go to Mesh/Vertex/ Make Vertex Parent. Come out of Edit mode, and the object should now move with the baked cached animation.
Edit: when I tried this yesterday, the object wouldn't move with the simulation if I used the menu item to parent it to a vertex on the cloth, until I cached the simulation and baked it. But now I find that caching and baking isn't necessary if I use Control P instead of the menu item. A bug I suppose.
Edit 2: If your object is elongated, you'll no doubt have a problem with orientation. In that case, you can place an empty at both ends of the object, make the empties into children of the nearest cloth vertices, then "hook" vertices on each end of your object to their respective Empty.
The hook command is under Mesh/Vertices/Hook/Hook to selected object - you have to have, in object mode, selected the empty first, then the object, then go into Edit mode so you have access to the objects vertices. The object will need all vertices parented to the one or the other Empty.