The League Two side gave as good as they received in a defence-dominated affair before falling behind to a Chris Shuker strike on 69 minutes.
Ian Moore came close to putting Tranmere in front in the fourth minute but his excellent chip over keeper Kenny Arthur from the edge of the box rebounded off the bar.
There was a setback for Stanley three minutes later when Robbie Williams was helped off with an ankle injury and Colin Murdock took over in defence.
Paul Mullen could not connect properly with a dangerous cross by James Ryan in the tenth minute with home keeper Danny Coyne grateful to gather possession.
Coyne had to be at his best two minutes later as he turned aside a well-struck shot by John Mullin.
The Stanley skipper was booked shortly afterwards for a challenge on Bas Savage.
Steve Jennings volleyed wide after some good Tranmere approach work and Moore was off-target with another firm strike.
The visitors were giving as good as they received at this stage and Murdock was only just wide with a back heel flick following a free-kick just after the half-hour.
Tranmere brought on Ed Sonko for Gareth Edds and changed to a more attacking formation immediately after half-time, but defences continued to dominate and the hour-mark passed without another shot on target.
Ryan almost broke the deadlock on 67 minutes but his free-kick from outside the corner of the box hit a post.
Two minutes later, Tranmere were ahead with Steve Jennings setting up Shuker for an angled shot past Arthur.
Tranmere almost had a second in the last minute but Arthur saved well from a close range effort by sub Chris Greenacre.