The Seagulls struck the crossbar through defender James Tunnicliffe before Charlton broke the deadlock against the run of play when Burton got in between two defenders to head home a free-kick by Nicky Bailey in the 29th minute.
Midfielder Wagstaff celebrated his first league start for Charlton by doubling their advantage eight minutes later from six yards after defender Grant Basey drilled the ball in from the by-line.
Brighton goalkeeper Graeme Smith prevented a heavier defeat for the Seagulls late on by producing a brilliant double save to deny striker Dave Mooney.
Brighton bossed the game before Charlton scored and they almost went in front in the 17th minute when Tunnicliffe was denied by the woodwork with a header following a corner to the near post by Elliott Bennett.
Christian Dailly missed a chance to extend Charlton's lead at the far post shortly after the break, while Brighton had another let-off when Mooney was off target at the near post following a corner by Elliot Omozusi.