Just to put a little more theory to this discussion -
in the subthreshold region, source/drain current in a MOSFET is described by the following functional relationship with threshold voltage Vt:
Ids ~ exp[e(Vg-Vt)/(mkT)] = exp[(Vg-Vt)/(mkT/e)]
where e is electron charge, Vg is the gate voltage, Vt is the threshold voltage, k is the Boltzmann's constant, T is the absolute temperature, and m is the body effect coefficient (dimensionless constant usually larger than 1.0, more specifically - m=1+Cdm/Cox; devices with m value below 1.0, also called as having a steep sub-threshold slope, are a subject of intense research in the recent and coming years).
"Better electrostatic control" of the gate over the channel means smaller value of m.
For smaller value of m (a derivative of this parameter - S=2.3*(kT/e)*m - is called subthreshold slope), you can allow for a lower Vt by keeping the leakage current (Ids) the same. And lower Vt will allow you a higher saturation current (current at high Vg, above Vt).
Hope this explanation helps to understand the relationship between Vt, subthreshold slope, and leakage current - from the theory / physics viewpoint...
All these formulas, and more, can be found in any textbook on MOSFET or semiconductor device physics - Sze, Taur and Ning, etc.