The word you’re looking for is simply “mème”.

The word, meme, used in the expression “Internet meme“, is a short form of the Greek word mimēma; the French word is identical except for the accented è:

mème Internet

Définition
Élément culturel propagé de façon virale sur le Web.

Notes
Le mème Internet peut notamment prendre la forme d’une
vidéo, d’un site Internet, d’un mot, d’une phrase, d’un personnage ou
d’un phénomène.

mème Internet n. m. […]

[ Le grand dictionnaire terminologique (GDT) – mème Internet ]

See also: Termium, Wikipédia. Note the word’s absence from FranceTerme, so there may indeed not be a set usage. On the other hand, as someone pointed out, it is not necessarily the first time such words have been used(see for instance this and this). The mème Internet is seen as a specific instance of the word mème; it is functionally distinct from Dawkins’ original idea.¹ Surely many people would understand you if you asked: “Tu connais le (mème Internet du) Harlem Shake/doge ?”.

_{1. When the term is not used, the “it went viral” idiom will often be leveraged with verbs describing a change of state, such as with “ce clip est devenu viral – un clip viral” (has become; could be used with blague, chanson, image etc., but not as a technique it seems). Similarly, you may find expressions such as “ça c’est/un clip qui s’est… répandu/propagé comme une traînée de poudre” (which spread like wildfire); depending on context it could simply be summed up in a familiar way as “un truc accrocheur, populaire, à la page, un phénomène, qu’on voit partout dans Internet ou sur le Web/la Toile et donc que « tous » connaissent.”.}

It is also often said using the english prononciation for the first e: mime

This is used to underline the comic action, just like a mime artist do.