Davies showed bravery and agility to keep out two David Clarkson shots and a volley from Oliver Norburn as Rovers turned up the heat following a sterile first half.
Clarkson should have scored after 64 minutes when Fabian Broghammer's cross found him in yards of space, but his fierce shot was too close to Davies who beat it away.
Neither goalkeeper had to make a save in a low key first half littered with passing errors. Broghammer twice went close for Rovers with a header and a shot just over, while Joe Anyinsah looked dangerous at times.
Fleetwood were even less effective going forward and an early shot from long range by Andy Mangan which flew over was the nearest they came to troubling debutant Rovers goalkeeper Neil Etheridge, who on loan from Fulham.
The visitors failed to create a clear-cut chance and had to work hard defensively for their point.