The visitors then threatened at the other end, Nazon attempting an overhead kick from just inside the box, but the ball landed on the roof of the net.
Oldham, who are hovering just above the League One relegation zone, failed to capitalise on a number of half-chances and Walsall made the most of their reprieve.
Fitzwater put the Saddlers in front shortly after the half-hour mark by nodding home the rebound after Placide had kept out Luke Leahy's initial header.
However, Richie Wellens's charges drew level on the stroke of half time through Nazon, who took the ball around two defenders and slammed into the bottom corner of the net.
The hosts continued to impress in the second half and had several opportunities to win the game, with Nazon forcing a good save from Liam Roberts.
Ultimately, however, both sides had to settle for a point at Boundary Park, with Oldham now three points clear of the dropzone and Walsall boasting a six-point cushion above the bottom four.
OLDHAM ATHLETIC (4-3-3): Placide; Moimbe, Gerrard, Edmundson, Hunt; Fane, Gardner, Byrne (Davies 90'); McEleney (Obadeyi 75'), Nazon (Holloway 69'), Doyle
WALSALL (4-4-2): Roberts; Leahy, Fitzwater, Guthrie, Devlin; Chambers, Morris, Dobson, Kinsella; Ngoy (Shaibu 68'), Bakayoko