Cowdenbeath's Gareth Wardlaw slotting into the bottom corner for the reply, before Paul McManus missed a penalty.
Young thought he had snatched the points for the hosts with 81 minutes played, but Wardlaw bagged his second with an 18-yard drive.
East Fife: Brown, Watson, Fagan, Smart, Ovenstone, Muir, Young, McCunnie, Linn (Conway 85), McManus, Crawford. Subs Not Used: Stobie, Sludden, Cargill, Kerr.
Goals: Linn 15, Young 81.
Cowdenbeath: Hay, Armstrong, Baxter, Mbu (Droudge 25), Linton, Fairbairn, McQuade (Ferguson 77), Winter, McBride (Ramsay 63), Wardlaw, Robertson.
Subs Not Used: Bower, Stein.
Goals: Wardlaw 31, 87.
Ref: J McKendrick
Source: BBC Sport