Maybe it 'sounds wrong' to you when you hear 'one dog were', but you must consider 'two cats and one dog' to be a plural number (three), of animals, and they were sitting on the fence. A cat and a dog are fighting. Two apples and one orange are on the table. Six dogs, eight cats, three men, two women, and one baby were in the room. In any case, if you didn't like it, you could always write or say 'one dog and two cats were...' and preserve the meaning.
Most constructions using 'and' of the form something and something else must refer to more than one thing (a plural quantity), and a plural verb must follow. One exception might be the name of a company or group considered as one thing, for example Brown and Root is an American company.