The teams were tied at 17-17 heading into the very late stages, after four penalties by Ignacio Mieres and a try from Sireli Naqelevuki for Exeter.
Extra-time seemed likely, but Williams pounced for the decisive try in the 79th minute to shatter the visitors.
The first points arrived in the 21st minute when Stade Francais were penalised for going over the top and Argentinian fly-half Mieres slotted a 15-metre penalty between the uprights.
Stade Francais were reduced to 14 men when their own Argentinian fly-half, Felipe Contepomi, was yellow-carded for fighting on the floor with Chiefs hooker Chris Whitehead and throwing a punch.
Mieres ignored the jeers of the French supporters to put the Chiefs further ahead with his second penalty, only for scrum-half Julien Dupuy to respond with a 40-metre kick after a high tackle.
Mieres landed his third penalty before Stade Francais, now restored to full strength, hit back. Dupuy kicked his second penalty on the stroke of half-time, leaving Exeter 9-6 in front.
The Parisians started the second half strongly and Contepomi put a neat grubber kick through for full-back Djibril Camara who collected the awkward bouncing ball and dived over the line to put the home side ahead.
Dupuy was off target with his conversion attempt from the left touchline but then he added penalties either side of one from Mieres, meaning the Chiefs were five points adrift going into the final quarter.
That lead was wiped out though as Stade Francais prop Rabah Slimani received treatment. The game continued, and replacement Naqelevuki shrugged off a tackle to power over the line to level the scores in the 67th minute but Stade Francais made a break led by Dupuy that put in Williams for the killer try.