Stephens, a cultured midfielder who was briefly linked with a move to German giants Bayern Munich in the summer before plumping for the Valley, swept home to give his side a deserved lead midway through the first half.
The former Oldham playmaker had a hand in the game's outstanding moment with 50 minutes gone, finding Scott Wagstaff who thrashed a volley into the net from 25 yards.
Johnnie Jackson scored 15 goals from midfield for the Londoners last season and he got up and running for the current campaign, netting a penalty with 13 minutes remaining.
Bournemouth failed to offer much in the way of an attacking threat, and Lee Bradbury's side were some way short of the form that took them to the play-off semi-finals last term.
Marc Pugh had their best chance after 56 minutes but shot straight at Charlton goalkeeper Rob Elliot.