Performing local extension from pseudoconcave boundaries along Levi-Hartogs figures and building a Morse-theoretical frame for the global control of monodromy, we establish a version of the Hartogs extension theorem which is valid in singular complex spaces (and currently not available by means of (partial derivative) over bar techniques), namely: For every domain Ω of an (n - 1)-complete normal complex space of pure dimension n >= 2, and for every compact set K subset of- Ω such that Ω\\K is connected, holomorphic or meromorphic functions in Ω\\K extend holomorphically or meromorphically to Ω. Assuming that X is reduced and globally irreducible, but not necessarily normal, and that the regular part [Ω\\K](reg) is connected, we also show that meromorphic functions on Ω\\K extend meromorphically to Ω.