Charlton had a 14th-minute penalty claim ignored when Gary Doherty was hauled back in the area by Graeme Lee in Saturday's League One clash at The Valley.
Notts County came close twice in quick succession after 33 minutes when Kevin Smith's header was deflected behind for a corner by Doherty's head and then Jon Harley's 30-yard shot was pushed behind by Rob Elliot moments later.
Lee Hughes should have broken the deadlock on the stroke of half-time but his volley went over the bar.
Matt Fry tripped Davies in the 74th minute as he broke into the area, an offence which earned County a penalty and the Addicks defender a yellow card.
But Elliot used his legs to save Lee Hughes' penalty and County were made to pay for their wastefulness five minutes from time when substitute Anyinsah - on his debut - fired a low shot past Stuart Nelson from 10 yards.