"Fly with" is definitely not incorrect.
To my ear "fly with one engine" is slightly more formal than "fly on one engine", and if I were writing a scholarly essay, I'd opt for "fly with" - but that's just a feeling; I've never written and barely ever read scholarly papers on aviation.
Prepositions like with and on have many and varied uses, so it's rather difficult to guess which one will be idiomatic, and probably pointless to try to translate directly from one language to another.
In this case, with probably means either
6 a —used as a function word to indicate the means, cause, agent, or instrumentality
As in "the dolphin can swim up to 60 kilometres per hour with its powerful tail fin"
Or
4 a —used as a function word to indicate combination, accompaniment, presence, or addition
As in "I went hiking with my new backpack"
While on might mean
2 b —used as a function word to indicate a source of dependence
As in "this car runs on electricity"
Or a definition that I can't find in a dictionary, but which we can see in examples like, "I bought fifteen bottles of wine at the chateau, but they were so good, I'm already on my last bottle."