When you think about the thrill of victory and the agony of defeat, associate a rout with the agonizing defeat. A rout is the kind of humiliating loss that makes you wish you would have been injured in the first quarter so you could have avoided the outcome.
One way to remember the meaning of the word rout is to consider its similarity to the word riot, both suggesting a disastrous mess. One interesting thing about the word rout is that whether it's used as a noun or as a verb, it's spelled the same and implies a similar meaning. "The losing team suffers during a rout while the winning team actually routs them!" Whichever form you use, you're referring to a pretty devastating defeat.