This is how I added css file. CSS file is in public/css/style.css.
<link rel="stylesheet" href="{{ URL::asset('css/style.css') }}">
Disabling the *.css.map functionality at gulpfile.js
At last, Solved the issue. Reason behind the problem is, for each file that is compiled, a companion *.css.map file in the same directory by elixir. And the *.css.map file was redirecting the sass file in stdin file, that made the php server confused, which css to load. I stopped the file from being automatically created by putting the line in gulpfile.js before the elixir mix function.
elixir.config.sourcemaps = false;