Kari Arnason put the Millers ahead in the first half, driving home a loose ball after Davies had dropped Mark Bradley's looping free kick, but the Cod Army's number one more than made up for his error after the break.
Jon Parkin had volleyed home Shaun Beeley's cross to equalise as the lead was shortlived, before Davies stole the show with a string of fine saves.
After injury forced the Millers into a double change at the break with Arnason and winger Jamie Devitt going off, Alex Revell and substitute Gareth Evans were denied by a double save from point blank range at the start of the half.
Kayode Odejayi then set up Revell again who's weak effort was saved before Davies pulled off his best stop of the evening, diving away to his left to tip away a Revell header with 12 minutes left to leave the visitors frustrated.
The home side's best effort came late on as Dean Howell's low drive was easily saved by Andy Warrington.