Stanley, who have now lost 14 games on the home soil this season, still need a point to secure survival after another poor display.
Barnet forced all the pressure in the first half as Adam Birchall fired over in the opening stages while Jason Puncheon hit the post with a free-kick.
The Bees took a deserved lead in the 53rd minute when Birchall crossed and Neal Bishop headed home.
Accrington’s Shaun Whalley then saw his deflected effort hit the crossbar, while goal-hero Bishop was stretchered off after trying to clear the rebound.
Barnet continued to be the major threat and Puncheon’s corner nine minutes from time found Michael Leary – who headed home his first goal of the season.