Was there a generic toast that Romans would say to each other when drinking, along the lines of Cheers, or Sláinte. It doesn't need to have the meaning of those so much as have the same cultural usage: something you say to another when toasting, or even just to yourself.
Here's a partial answer:
Prosit (lit. may it be useful) came to mind (as I commented), but I couldn't easily find it attested. The fact that it is used in languages other than German suggests a former widespread culture of using prosit in toasts that could perfectly come from a time when Latin was the cultured lingua franca. But it could as well be the surviving word of a longer formula that was used just in modern Latin.
Now, an unsourced WP article states that there are two words for a toast in Latin, namely prosit and the verb propino.
The good news is that propino is well attested and means To drink to one's health, to pledge (apparently this pledge refers to the archaic meaning, i.e., to toast). It is even attested in the context of a toast (apparently, you judge):
talos arripio, invoco almam meam nutricem Hérculem,
iacto basilicum; propino magnum poclum: ille ebibit,
caput deponit, condormiscit. (Pl. Cur. 2.3.79:81)
I took up the dice, and invoked Hercules as my genial patron; I threw a first-rate cast, and pledged him in a bumping cup; in return he drank it off, reclined his head, and fell fast asleep. (Riley, 1912)
The downside is I have no proof that you can use propino (I pledge) alone as an interjection, and it is possible that you just can't. Instead, the safe way to use the verb from this example is to say 'I toast to someone' using the accusative (as in the example) or dative case.
3. A full, attested sentence
Another idea comes from this passage:
In case you want to use a full sentence, that is attested in the ancient literature as a formula for toasting (to one person) this is your way to go. Its plural form would be propino vobis salutem plenis faucibus (to more than one person).
4. salutem (?)
Now, wishing health (as in the last example) is the standard Latin way to wish someone well. So I think salutem alone could work (it works as a greeting, anyway). I don't expect it to be attested (and looking it up may prove frustrating), but the fact that nearly every romance language use the derived word to toast is consistent with it being used in some late stage of Latin.