Substitute Carcary broke the deadlock 13 minutes from time when he fired home from 12 yards.
Carcary also hit the post before Dumbarton's David Nugent was sent off in injury time for violent conduct.
Dumbarton: White, Nugent, Chisholm, McManus, McNiff, Tony Wallace, McStay (Geggan 69), Chaplain, Stirling Smith, McShane (Carcary 63), Campbell (Gary Smith 85). Subs Not Used: Winslow, Gordon.
Sent Off: Nugent (90). Booked: Tony Wallace, Chaplain, Nugent, Chisholm, McManus.
Goals: Carcary 77.
Stenhousemuir: Brown, Gibb, Thom, Smith, Stephen Thomson, Anderson, Clark (Quinn 79), Iain Thomson, Murray, Stirling, Dalziel. Subs Not Used: McLennan, Scuillion, Motion, Hunter.
Booked: Clark, Dalziel, Stephen Thomson, Gibb.
Att: 600
Ref: C Charleston
Source: BBC Sport
Source: BBC Sport