There was little argument against the decision with Alan Rattray clearly catching Linn on the edge of the area.
It was a tight game with neither team playing great football. The best chance of the opening 45 came on the stroke of half-time when Arbroath trialist David Winters turned his defender and was unlucky to see his half-volley bounce back off the upright.
The visitors started the second half stronger and David Muir should have scored when he headed well over from a free-kick.
Prior to Linn's winner, the home side forced a succession of corners which came to nothing.
Arbroath: Hill, McMullan, Rennie, Rattray, McCulloch, Doris, Gibson, Lunan, Winters (McLean 61), Hislop, Jackson (Scott 77). Subs not used: Morrison, Moffat, Suttie.
East Fife: Brown, Watson, Smart, Ovenstone, S Campbell (R Campbell 83), Muir, McCunnie, Young, McManus (Cargill 90), Crawford (Conway 77), Linn. Subs not used: Baillie, Cargill, Conway, Fagan.
Att: 376 Ref: F McDermott