The hosts were two up just before half-time when Connor Gethans swept in a cross from Daniel Moore.
Four minutes after the interval Wyness was teed up by David Donald and drilled in the third, while Scott Dalziel added a consolation goal for Stenhousemuir.
Peterhead: Bateman, Donald, MacDonald, Stuart Smith, Moore, McVitie (Strachan 88), Anderson, Emslie (Clark 70), Bavidge, Wyness, Gethans. Subs Not Used: Wilson, Scott Ross, Hepburn.
Booked: Moore. Goals: Wyness 9, Gethans 43, Wyness 50.
Stenhousemuir: Brown, Lyle, Fusco, Smith, Stephen Thomson, Anderson (Scuillion 61), Iain Thomson, Clark (Stirling 70), Murray, Dalziel, Williams. Subs Not Used: McLennan, Motion, Quinn. Booked: Williams.
Goal: Dalziel 76.
Att: 442
Ref: B Colvin
Source: BBC Sport
Source: BBC Sport