Jeremy Crawford unofficially tweeted that this works for wands, though the rules do not mention it.
Since the only hint (apart from their names) that magical rods, wands and staves can be used as foci are their tags rod, wand and staff that classify them as such, then we look at their tags descriptions at pages 139-140 of DMG but as you previously said, there is no explicit information about their use as arcane foci.
However, according to Jeremy Crawford, any wand can be used as an arcane focus. I can't see a reason why it wouldn't also apply for rods and staves since they usually are also foci but none of them had it written on their descriptions.
@JeremyECrawford Can Wand of the War mage be an arcane focus? Doesn't specify, and while wands can be foci, it's not clear that ANY wand can
JC: Any wand can be used as an arcane focus.
Jeremy Crawford tweets are no longer official rulings; only those in the Sage Advice Compendium are. However, the SAC does not provide satisfactory answer on this subject, so Crawford's tweet does indicate the designer's opinion on the matter, for what is worth.
The descriptions of DMG magic items are not exhaustive.
An arcane focus is defined as "a special item ... designed to channel the power of arcane spells". Magic items of the "rod" category are "typically made of metal, wood, or bone ... about 2 or 3 feet long, 1 inch thick, and 2 to 5 pounds." Not all mundane rods of those dimensions are arcane foci (some may be, for example, cart axles, or crowbars); thus not all magic rods are. In contrast, all swords are weapons, by definition; it's not a special subcategory of swords that are made for attacking
First, I wouldn't call magic rods and staves something mundane; they are magical.
Second, the DMG magic items descriptions (p139-140) don't describe their every function, they do not specify their most basic functions since magic wands have nothing written there that they can be used as foci as weapons doesn't say that they can be used to attack. The same thing about rods and staves that doesn't explicitly say about their use as foci. RAW is the same to say that I can use a normal sword to cut someone but I can't use a Dancing Sword to do the same because isn't written that I can unless I make it fly while in my hand it doesn't have the capability to injure anyone. That would be unreasonable.
PHB also doesn't help.
There are no non-foci rods or wands described in the PHB. Would all rods and wands in existence be foci? That would also be unreasonable since it clearly says they are specially prepared to be used as such. If you imply that someone could make a non-foci rod with a log to support a broken table or a non-magical wand with silver and jade for a teacher to point at the blackboard because it's reasonable but isn't written in the book, then someone can make an ornamental sword that isn't a weapon and also isn't written there. Just because isn't written in the book doesn't mean it can't be done.
Marq said in his answer:
Magic items that can be used as a spellcasting focus, such as a hat of wizardry, say so explicitly:
This antiquated, cone-shaped hat is adorned with gold crescent moons and stars. While you are wearing it, you gain the following benefits:
- You can use the hat as a spellcasting focus for your wizard spells.
Unlike wands, staves and rods, hats are not usually arcane foci. If a specific item that usually doesn't have such function now works as such, obviously it need to be stated in its description. That's the reason the hat of wizardry clearly have it noted there. Since wands, rods and staves are meant to be used as foci as much as weapons are meant to be used to attack, they didn't need to have it noted in their entries; it's their basic function, you just assume they do what they are supposed to do.