Player-boss Davids saw red with five minutes left after two bookings in quick succession, the second for a foul on Stanley's new player-coach James Beattie.
Stanley went ahead seven minutes after half-time when Romuald Boco marked his 100th appearance for the club with his sixth goal of the season.
Bees goalkeeper Graham Stack blocked his initial shot but Boco made no mistake with his second attempt.
They could have doubled their lead on the hour but James Gray headed over.
Instead Barnet were awarded a penalty with 20 minutes to go and although Mark Byrne's first effort was saved by Ian Dunbavin, the referee ordered a retake after deeming the goalkeeper had moved off his line too early and the Bees midfielder made no mistake with his second attempt.