How to call a method dynamically in Elixir, by specifying both module and method name?

You can use apply/3 which is just a wrapper around :erlang.apply/3. It simply invokes the given function from the module with an array of arguments. Since you are passing arguments as the module and function names you can use variables.

apply(:lists, :nth, [1, [1,2,3]])
apply(module_name, method_name, [1, array])

If you want to understand more about how elixir handles function calls (and everything else) you should take a look at quote and unquote.

contents = quote do: unquote(module_name).unquote(method_name)(1, unquote(array))

which returns the homoiconic representation of the function call.

{{:.,0,[:lists,:nth]},0,[1,[1,2,3]]}

You can unquote the quoted function call with Code.eval_quoted/3

{value, binding} = Code.eval_quoted(contents)

Edit: here is an example using Enum.fetch along with a var.

quoted_fetch = quote do: Enum.fetch([1,2,3], var!(item));             
{value, binding} = Code.eval_quoted(quoted_fetch, [item: 2])