I am looking for the closest equivalents of the following phrase in Latin:
Don't get mad, get even.
Preferably not a word-by-word translation, but an 'established' phrase or proverb.
I'm not aware of an established proverb, but if you want to convey the meaning "don't become angry, but [calmly] avenge yourself" you could translate noli irasci, ulciscere or (to keep the parallelism of the English phrase) ne irascere, ulciscere.