Classification of surfaces using point set topology The classification theorem for surfaces is proved using triangulation of surfaces. Another way to do this is using Morse theory. 
Can the classification of surfaces be done using only point set topology without using triangulation or Morse function? 
 A: It is hard to prove negative as absence of evidence is not evidence of absence. All complete proofs of classification of compact topological surfaces that I know follow the following lines:
Step 1. Proving that every topological surface admits either a triangulation or a smooth structure. This is quite hard and does resemble point-set topology.  
Step 2. Proving that every triangulated surface is PL homeomorphic to one of the "standard models" (say, in the case of closed connected surfaces, a connected sum of tori and projective planes). For this, there are several strategies. Or working with smooth surfaces and use, say, Morse theory, as you mentioned. Here, hardly anybody bothers proving that every topological surface admits a smooth structure. 
Step 3. Proving that Euler characteristic is a topological invariant, thereby concluding the proof of the classification theorem. (This step might precede 1 and requires some, at least rudimentary, algebraic topology.)  
I am unaware of any textbook treatment of Step 1, going from topological to smooth surfaces. The textbook proofs I know proceed by constructing a triangulation (Rado's Theorem) or simply work with smooth surfaces to begin with. 
One reference for the existence of a triangulation is given here, Thomassen's paper. See more references in this Mathoverflow discussion. 
One place which has both the proof of Rado's theorem and classification of compact surfaces is Moise's book "Geometric topology in dimensions 2 and 3". 
Thus, assuming that you want to see the complete proof of classification of topological surfaces, take a look at Moise's book. Or start with  Thomassen's paper and then read any account of classification of triangulated compact surfaces. Or, follow the book (aimed at undergraduate students) 
Jean Gallier & Dianna Xu, A Guide to the Classification Theorem for Compact Surfaces, Springer-Verlag, 2013.
which also proves both Rado's theorem and classification of surfaces. 
PS. What's lost in the combinatorial proofs of classification of surfaces is the following:

Every topological (or triangulated) surface admits a unique smooth structure up to diffeomorphism. Moreover, every homeomorphism between smooth surfaces is isotopic to a diffeomorphism. 

This uniqueness does not automatically follow from the combinatorial classification of surfaces even in the compact case. The reference I know is 
J. Munkres, Obstructions to the smoothing of piecewise-differentiable homeomorphisms. Ann. of Math. (2) 72 (1960), 521–554. 
