The man, who has not been identified, was taken to hospital after being treated at the scene.
"We regret that this incident occurred. Our thoughts and prayers are with the contractor and his family," refinery manager Mark Northcutt told WAFB.
An investigation has been launched to look into the incident.
ExxonMobil says on the web page for its Baton Rouge area operations that "the safety of our employees, contractors and community comes first in all we do".
"We have a simple goal when it comes to safety - ‘nobody gets hurt'. By operating with responsibility and integrity, we are also working to find ways to reduce greenhouse gas emissions while operating efficiently and reliably to better meet the world's growing energy needs," the page says.
ExxonMobil is the biggest private employer in the parish and the second largest manufacturing employer in Louisiana, with about 300 employees and roughly 2000 additional contractors.

