update: My application has already been accepted! Thanks to everyone who has provided me valuable suggestions. I really appreciate your help! This is an important lesson to me: "Don't leave your work until the last minute"
I just missed the my grad school application deadline....... This is the biggest and most serious mistake I ever made in my life...
This university locates in the Central timezone so I assumed its application deadline was 11:59pm CST. Before I submitted my application, I rechecked everything out of a habit. Then I clicked to submit my application at 11:10pm CST but found that it couldn't be submitted because I was unable to choose the term of enrolment. I first thought it was an internet or technical problem so I restarted my search engine a few times but it still didn't work. At that time, I realized that the deadline might be in EST! (confirmed, as I could still choose term of enrollment for my application to another department in the same university whose deadline is at a later time).
I already sent the department with screenshot of my application with time displayed on the side, along with my application materials. I will also be calling the department first thing in the morning tomorrow to make an appeal. Taking the advice provided by the folks who responded to my question, I am also going to contact my prospective advisor for help.
Is there anything else I can do?