You might think that the 3 Stooges were running things ...
US Air Force Secretary Michael Wynne was "very concerned"
The shipment, which should have been helicopter batteries, was made in 2006 but only discovered last week.
The Pentagon says no nuclear materials were shipped and the parts have been returned to the US.
The issue of US arms sales to Taiwan is sensitive as China regards the island as a renegade province.
The BBC's Jane O'Brien in Washington says senior officials have described the incident as "disconcerting" and "intolerable".
Taiwan had pointed out the error, but owing to a two-year miscommunication the US administration remained unaware of it until last week.
The shipment had been sent from a US airbase in Wyoming.
Our correspondent says the components are not in themselves nuclear material but do form part of a long-range missile system that could deliver a nuclear weapon.
Air Force Secretary Michael Wynne said: "It is a component for the fuse in the nose cone for a nuclear system. We are very concerned about it."
President George W Bush and the Chinese government were informed about the error.
Beijing vehemently opposes US arms sales to Taiwan and has threatened to attack the island if it declares independence.
The mistaken shipment is the second blunder in recent months.
Last August a B-52 bomber flew across several US states mistakenly armed with nuclear-tipped cruise missiles.