The result they use is Moise's theorem:

Moise, Edwin E. (1952), Affine structures in 3-manifolds. V. The triangulation theorem and Hauptvermutung, Annals of Mathematics. Second Series 56: 96–114,

This states that a 3-manifold has a unique PL structure, and a unique smooth structure. 

Moise's theorem was considered so standard back then I imagine Gordon and Luecke didn't bother to cite it because it was so omni-present. 

So the idea goes like this, your homeomorphism can be promoted to a PL-homeomorphism of the sphere.  Since it is orientation-preserving you can apply the Alexander trick to isotope your PL homeo to the identity (through PL-homeomorphisms).  

You could state this as the theorem that the group of orientation-preserving PL homeomorphisms of $S^3$ is connected.   This argument was considered standard back then, too. 

If you want to go one further step as Misha suggests, you have to smooth your manifold.  Moise's theorem covers that step.  But proving that the group of orientation-preserving diffeomorphisms of $S^3$ is connected is harder.  That's Cerf's theorem. 

J.Cerf, Sur les difféomorphismes de la sphère de dimension trois (Γ4=0), Lecture Notes in Mathematics, No. 53. Springer-Verlag, Berlin-New York 1968.

Although Hatcher proved more (and you can in principle use his techniques to prove Cerf's result), Cerf's argument gives a very nice general technique.  It is the birthpoint of "Cerf Theory" meaning studying 1-parameter families of smooth functions, showing they can be assumed to be Morse at all but finitely-many times, and describing the cubic singularities where the family fails to be Morse.