There wasn't a set phrase for this as there is in English, but it can be expressed in a number of ways.
First, depending on exactly what part of sense you're going for, we could be talking about perception or disposition. I think more people think of it in terms of the latter, except when discussing someone's lack of humor. Funny enough, the two actually overlap in Latin. Words such as sensus and cogitatio would adequately cover both understanding of humor and disposition towards it, whereas something like perceptio or notio would chiefly refer to understanding.
For humor, we also have a variety of words at our disposal: festivitas, facetiae, lepor, and even sal all work and appear to be largely interchangeable. They all generally mean "wittiness", and can describe both a biting sarcasm (especially sal) or the farce of a Plautine play.
Do note though that in English 'humor' can mean two entirely separate things. In the phrase "sense of humor", it's referring to the comic, the amusing, the funny. But elsewhere 'humor' as a noun can mean state of mind, disposition, or temperament, as in the sentence "I'm in no humor for these shenanigans!" So when the dictionary lists 'humor' as a synonym for sensus, they intend to mean the latter, not the former.
Putting these together, you can essentially pick and choose which words you want to use. For what it's worth, Pope Francis in the exhortation luchonacho mentioned in his comment uses the phrase sensus leporis.
You could also combine two of the latter or even two of the former to really drive home the idea that you are discussing the idea of humor.
In general, though, I think a good option is: sensus festivitatis leporisque.
I would choose sensus because it forms the same model as sense of humor, and we see the same plasticity in the Latin: Lucretius talks about the sense of touch (2.435) and Cicero the sense of humanity (Verr. 2.1.47). Meanwhile, festivitas and lepor consistently connote a more genial humor than sal or facetiae.
However, perhaps a more encompassing turn of phrase could be: sensus festivitatis facetiarumque.
It's a bit more to say, but you get the jovial side in festivitas and the jovial and sarcastic side in facetiae. The latter word would be more appropriate for sarcastic and acerbus humor.
Two words together in this way better clarifies that we're talking about a larger concept (cf. phrases like mens animusque or mens ratioque), though at any point any one of the words could work alone with the sense intact, and the translation of the Pope's words are as good as any.