Nobody understands exactly how painkillers (aspirin, paracetamol etc.) work on the molecular level. Yet they are generally held to be quite useful.

Edit: electroconvulsive therapy is an even more extreme example; we have no clue how it works, but it's very effective on severe depression. We only know it works because some Italians back in the olden days, before ethics committees were invented, decided to electro-shock a bunch of "crazy people" to see what happened. The reason they tried it was that electroshocking of pigs for slaughter had been observed to give a temporary anaesthetic effect.