I'm looking for a single word that would describe the feeling of say, getting a happy ending at the end of a tragic movie—like 'glad', but stronger. Feels like it exists but I just can't recall it—but if it doesn't, is there a better way to describe it other than what I put in the title?
[…] the feeling of say, getting a happy ending at the end of a tragic movie--like 'glad', but stronger.
This feeling is called catharsis.
I don't think there's a word that fits into the same syntactic slot as glad, though. (There exists an adjective cathartic, but it's used to describe the experience — the thing that causes or triggers the catharsis — rather than the person.)
Euphoric comes to mind.
marked by a feeling of great happiness and excitement : characterized by, based on, or producing euphoria
I feel the word brings a "cleansing joy" aspect with it.
The word that comes to mind is 'joy':
Intense and especially ecstatic or exultant happiness, or an instance of such feeling.
[joy. (n.d.) American Heritage® Dictionary of the English Language, Fifth Edition. (2011). Retrieved November 3 2015 from http://www.thefreedictionary.com/joy.]
'Joy', however, does not denote relief; rather it denotes enormous satisfaction and contentment. In this way, 'joy' is properly esthetic: it is a static emotion. As James Joyce expresses it in his "Paris Notebook",
... joy is excited by whatever is substantial or accidental [Yale MS. adds after "accidental", "general or fortuitous"] in human fortunes ...
(From James Joyce: The Critical Writings, edited by Ellsworth Mason and Richard Ellman, Viking, 1959.)
Later in the same paragraph, Joyce adds that
... even tragic art may be said to participate in the nature of comic art so far as the possession of a work of tragic art (a tragedy) excites in us the feeling of joy. ... All art, again, is static for the feelings of terror and pity on the one hand and joy on the other hand are feelings which arrest us.
In a note on the text of the latter passage, the editors point out that "Joyce substitutes stasis for catharsis. That joy may result from tragedy as well as from comedy does not follow well from what he has said. In the Portrait he confines the discussion almost completely to tragedy and changes 'joy' to 'the luminous silent stasis of aesthetic pleasure.'"
Aside from quibbles about whether the emotion occasioned by tragedy is, properly, joy, or rather the unadorned "stasis of aesthetic pleasure" (an overwrought expression more suited to the developing character of Stephen Dedalus in A Portrait of the Artist as a Young Man), it is the stasis of contentment or satisfaction that constitutes such relief as may be felt in the apprehension of tragedy.
Perhaps gratification (the noun form of the adjective gratifying in @Mamta D's prior comment)
Pleasure, especially when gained from the satisfaction of a desire
Oxford Dictionaries Online