The 20-year-old goalkeeper was in fine form for the Cumbrians, in particular when he denied Ryan Lowe and then somehow Jake Cassidy, who had earlier hit the post, when he was clean through on goal.
Jean-Louis Akpa Akpro had a second-minute header cleared from underneath the crossbar while a Cassidy curler rolled agonisingly along the goal line after first striking the inside of a post.
Lowe saw his venomous volley turned away by Pickford before Brad Potts threatened the Tranmere goal from long range midway through the second half.
Pickford was at his best when he produced a superb reaction save to deny Cassidy after he looked destined to score following a neat pass by Akpa Akpro.
Tranmere wasted several decent chances to take net a winner as time ticked away with Cassidy, Akpa Akpro and substitute Gboly Ariyibi all guilty of missing the target.