Toulon took the lead through Armitage's superb score and they almost had another inside four minutes following more good work by the flanker, only for scrum-half Sebastien Tillous-Borde to drop the ball as he tried to offload from a tackle.
But poor tackling let Morgan Turinui through to set up full-back Hugo Bonneval for a converted score in the 12th minute which nudged Stade ahead.
Fly-half Jules Plisson kicked two penalties and a drop goal to extend the visitors' lead to 16-5 but Wilkinson was on target with penalties in the 32nd and 40th minutes, either side of Plisson's third, to leave the half-time score at 19-11 to Stade.
Wilkinson took his side into the lead with three penalties and a drop goal in the first 15 minutes of the second half but Turinui rounded off a wonderful passing move from the base of a scrum to restore the visitors' advantage, Plisson adding the extras to make it 26-23.
Two Wilkinson penalties countered by a Plisson drop goal left the scores level at 29-29 going into the final 10 minutes - but with Stade winger Julien Arias in the sin-bin for cynically preventing a try after Plisson's kick was charged down.
Wilkinson missed the resulting penalty and then a drop-goal attempt before remarkably putting a 30-metre effort wide from a central position - but he kicked the crucial drop goal from just over 40 metres a minute from time to ensure his side will face Biarritz or Brive in the May 18 final.