The meanings of the terms are indeed close, so it is quite understandable that consulting the dictionaries left the OP puzzled as to what the difference is.
The principal difference between them is not in the meanings that would be captured in the explicit definitions of the kind that can be found in the dictionaries, but in their being anchored in very different fields of discourse. While the word synergy has been around for a while, in the recent decades it became an established term (some would say a buzzword) in the field of business, and that's where most of its present-day use is. Using the term (particularly using it frequently) thus tends to identify the speaker as business-school educated (unless the context has to do with medicine or related sciences). Using gestalt, on the other hand, tends to identify the speaker as somebody who was educated in psychology or philosophy. There is practically never an occasion for both words to be used within the same text, so the question of how they are related to each other never arises.
In so far as there is a difference in meaning between the two, it is that synergy involves looking at the matter from the perspective of the parts, while gestalt involves looking at it from the perspective of the whole. One can thus say that there is synergy between A and B, when these two things, which one has first identified separately, work together to produce something that is more than their sum. On the other hand, if something is a gestalt, one starts by perceiving it as a whole; it is only later, and with some work, that one may be able to discern what its parts are, and come to realise that it is more than their sum. The term gestalt was introduced into English in the context of Gestalt psychology, so some familiarity with it will help one to get the sense of how to use the word competently.