Create a Jenkins job, ( Create a free-style software project)
In the configuration of that job there is a "Build" section with "Add build step" pulldown.
Add a "Execute Shell" step, and insert your script path.
sudo sh /path/to/script
Allow jenkins to run the script in /etc/sudoers
jenkins ALL = NOPASSWD: /path/to/script
jenkins ALL=(ALL) NOPASSWD:ALL
jenkins ALL=(ALL) ALL
Now run the Jenkin Job
Hope this helps!