Quasi-equilibrium is different from classical chemical equilibrium, but can be described using a similar thermodynamic treatment.  Consider the reaction below
where complete equilibrium is achieved between all the species in the system including activated complexes, [AB] .  Using statistical mechanics, concentration of [AB] can be calculated in terms of the concentration of A and B.
TST assumes that even when the reactants and products are not in equilibrium with each other, the activated complexes are in quasi-equilibrium with the reactants. As illustrated in Figure 2, at any instant of time, there are a few activated complexes, and some were reactant molecules in the immediate past, which are designated [AB] (since they are moving from left to right). The remainder of them were product molecules in the immediate past ([AB]).
In TST, it is assumed that the flux of activated complexes in the two directions are independent of each other. That is, if all the product molecules were suddenly removed from the reaction system, the flow of [AB] stops, but there is still a flow from left to right. Hence, to be technically correct, the reactants are in equilibrium only with [AB], the activated complexes that were reactants in the immediate past.