Nicky Weaver made two early saves to deny Matt Fry and Johnnie Jackson but there was nothing the former Addicks keeper could do to prevent Charlton going in front after 23 minutes.
Wagstaff saw his first attempt blocked but the rebound fell kindly to him on the edge of the area and the Addicks' top scorer placed a low shot into the bottom corner of the net.
Wagstaff was denied a second by Weaver in the 31st minute when his 20-yard volley was pushed behind.
Tommy Miller came close to creating an equaliser in the 63rd minute when his low delivery went across the face of goal but there was no Wednesday player to apply a final touch.
Owls substitute Clinton Morison had a penalty claim ignored after he appeared to be clipped by Charlton keeper Rob Elliot and then Neil Mellor failed to get his header on target after rising high to meet Jermaine Johnson's cross. Daniel Jones then blasted an 89th-minute free-kick wildly over the bar, a move which summed up a lacklustre display from Wednesday.