You can try and reverse how pull requests happen.
- Go to your fork
Issue a Pull Request
By default this will be your fork on the right (head repo) requesting to push its commits and changes to the original repo (base repo) on the left.
Click the drop down for both base repo and head repo and select each other's repos.
You want yours listed on the left (accepting changes) while the original repository is on the right (the one with changes to push). As illustrated in this image:

Send the pull request
If your fork has not had any changes, you should be able to automatically accept the merge.
If your code somehow conflicts or is not quite clean enough, then this will not work to update via the GitHub web interface and you will need grab the code and resolve any conflicts on your machine before pushing back to your fork.