With one minute remaining, Stanislas curled his free-kick into the top corner for a vital three points.
Jay Rodriguez had put West Brom ahead four minutes after the break, when he volleyed in Salomon Rondon's nod-back from a cross.
Jordon Ibe equalised from distance on 77 minutes with a low, bouncing effort that beat Ben Foster.
Stanislas's 89th-minute strike lifted AFC Bournemouth to 10th on 36 points.
A seventh successive defeat keeps West Brom bottom, on 20 points, 10 from safety.
AFC Bournemouth: 31 Mar v Watford (A)
West Brom: 31 Mar v Burnley (H)