In the novel the phrase is мир спасёт красота (mir spasyot krasota), "the world will be saved by beauty" (our translation); the proverbial form, красота спасёт мир, is also in the book, in Hippolyte's exclamation right after. The prince never says it before the reader: Hippolyte heard it from Kolya and repeats it with mockery, and Aglaya puts it on the list of what he must not say to the guests.
Where It Appears
Quotes
Is it true, Prince, that you once said that beauty will save the world? [...] And I maintain that he has these playful ideas because he is now in love.
Part Three: The Explanation, Hippolyte
if you start to talk about something like the death penalty, or the economic situation of Russia, or of "beauty will save the world," then... I, of course, will be delighted and laugh heartily, but... I warn you in advance: do not show yourself before my eyes afterward!