Let's define what is exactly contemporary satire. This genre belongs to a form of comedic literature whose purpose is to present a mocked scenario as it if were serious. This is an aim to criticize and basically laugh about serious topics that, in a contemporary satire, will be ridiculed and belittled.
In "The Rape of the Lock", Pope criticizes the aristocrats of his time with their ridiculous social rules and expectations by blowing out of proportion an argument between two rich families over a stolen lock of hair. This is what is known as the idiom "a tempest in a teacup", or the exaggerated rendition of a petty situation.
Pope modeled the family feuds as an epic battle almost identical to The Iliad, complete with a mirror version of the abduction of Helen of Troy in the form of the rape of the lock. Everything from excerpts of The Illiad are quoted, the names of gods are used for characters, and the exaggeration of it all is what makes it a satire.
If "The Rape of the Lock" were a traditional poem the situations would not be satirized nor mocked, especially to the extent that Alexander Pope uses it. The theme of love, beauty and courage are idealized in Pope's age. Hence, what he did is like a "mockumentary" of one of those uber romantic and emotional episodes to humor a population that craved a change of pace.