It seems indeed pretty clearly to have been **Riemann**, in [§6 of his Inaugural Dissertation](http://books.google.com/books?id=EzIPAAAAIAAJ&pg=PA9) (1851). There he defines *zusammenhängend* as well as *einfach*, *zweifach* and *mehrfach zusammenhängend* without citing any prior sources.

Yet a possible antecedent is **Gauss**, of whom J.-C. Pont's book (cited in the comments above and reviewed [here](http://www.persee.fr/web/revues/home/prescript/article/rhs_0151-4105_1976_num_29_1_1382)) singles out a text, written around 1840, "which one can consider as a sketch of the theory of the order of connectivity". It would be interesting to dig Pont's exact reference, which ought to be available in Gauss's [*Werke*](https://archive.org/search.php?query=Gauss%20Werke).

(On Gauss and Riemann see also a [letter of Betti](http://dx.doi.org/10.1007/BF00327626) published by A. Weil, *Riemann, Betti, and the Birth of Topology*, Archive for History of Exact Science **20** (1979) 91-96.)