In his address of the poem to the heroine, Miss Arabella Fermor, Pope tells why there is supernatural machinery in the poem:
The Machinery, Madam, is a Term invented by the Criticks, to signify that Part which the Deities, Angels, or Daemons, are made to act in a poem. . . .
Because Pope is writing a paradoy of a classical Greek epic poem, which always features the greater and lesser gods in the ancient Greek belief system, Pope has to populate "The Rape of the Lock" with similar gods and divine helpers to that the parody mirrors its source, the epic.
At the beginning of Canto 1, in fact, the poet assures the heroine, Belinda, that she is surrounded and constantly watched:
Know then, unnumber'd Spirits round thee fly,/the light Militia of the lower Sky;/These, tho' unseen, are ever on the Wing,/Hang o'er the Box, and hover round the Ring.
Pope establishes here that Belinda, like any Greek epic hero or heroine, is protected at all times by minor gods, in this case, he calls them "Sylphs," whose job is to make sure nothing befalls Belinda or her belongings. The humor (and irony) is that this supernatural machinery is operating only to protect a lock of hair rather than to protect an epic hero fighting for his own and his country's survival.
Belinda's chief protector is a Sylph named Ariel:
This Nymph, to the Destruction of Mankind,/Nourish'd two Locks, which graceful hung behind/In equal Curls. . . .
One of Pope's purposes for the poem is to gently remind the warring aristocrats that issue of the stolen lock of hair is, in relative terms, ridiculous, and so he carefully weaves this epic structure around the locks of hair--in the quote above, we have a supernatural being whose only job is to look after the locks of hair.
When, in Canto III, the poem reaches a critical point--the rape of the lock--Pope describes again how the supernatural machinery operates to try to save Belinda's hair. Just as Lord Petre is about to cut Belinda's lock, we read
Ev'n then, before the fatal Engine clos'd,/A wretched Sylph too fondly interpos'd;/Fate urg'd the Shears, and cut the Sylph in twain. . . .
The sacrifice of a minor supernatural being has close parallels in Greek epics, and we also must note that, because this is a mock epic, the action has to be at least partially the result of fate, not human action--"Fate urg'd the Shears" is simply another aspect of the supernatural machinery at work.
Pope's inclusion and use of the supernatural machinery is critical to the overall effect of this mock-epic poem: while deflating a few egos, and putting into perspective the loss of a lock of hair, the poem must reflect the conventions of classical Greek epics, and the supernatural elements are an indispensable conventional element.