Exactly. Initially, prices look good overseas in terms of exchange rates but as soon as we pick up production here in the US to meet demand prices actually go up because of the weaker dollar -- thus prices overseas need to be raised to offset the cost of production.
Add back in the fact that American consumers are the largest consumers in the world and you can see that the rest of the world will be affected by higher prices here.
In trade, when one entity is affected they all are affected. It might take some time to realize this in the market but eventually it catches on. For example, the reason why we are seeing the collapse of the housing market now is attributed to the credit expansion that happened from 2002 to 2006 under Greenspan. Planners are not able to foresee further into the future than the most immediate effects that some action might have. Bastiat warned against the unintended consequences of economic planning in "That Which Is Seen, That Which is Not Seen."
Here is a better explanation than I can give:
"In the department of economy, an act, a habit, an institution, a law, gives birth not only to an effect, but to a series of effects. Of these effects, the first only is immediate; it manifests itself simultaneously with its cause--it is seen. The others unfold in succession--they are not seen: it is well for us, if they are foreseen. Between a good and a bad economist this constitutes the whole difference -- the one takes account of the visible effect; the other takes account both of the effects which are seen, and also of those which it is necessary to foresee."