The South African midfielder lashed home a superb finish as rejuvenated Palace made it back-to-back wins - having previously lost their opening three matches.
They also leapfrogged their newly-promoted neighbours, who have now lost two on the bounce, in the Championship table.
Charlton will feel hard done by, though, not least because Bradley Wright-Phillips had what looked a perfectly good goal ruled out for offside.
They also had chances following Dikgacoi's decisive strike just after half-time, but Palace held on to take the points the short journey back to Selhurst Park.
Palace could have been two up in the opening 10 minutes with Wilfried Zaha tormenting Charlton's defence early on.
Twice the teenage winger's strength, pace and skill took him to the byline but Dikgacoi blazed his first cross over. Moments later another powerful, weaving run and cross found Glenn Murray, whose effort was blocked by Addicks keeper Ben Hamer and Leon Cort hacked Owen Garvan's follow-up off the line.
But Charlton were denied an opening goal on the half-hour when Cort nodded Lawrie Wilson's free-kick to Wright-Phillips.
The striker, watched by father and Palace legend Ian Wright in the stands, appeared to be onside as he buried a header past Julian Speroni, but the linesman's flag cut his celebrations short.
The hosts were well on top as the first half came to a close with Yann Kermorgant heading Johnnie Jackson's cross wide and Dale Stephens, who saw a deadline-day move to Aston Villa fall through, driving narrowly over.
But it was Palace who took the lead five minutes after the interval when Charlton failed to deal with Garvan's corner.
Damien Delaney was unmarked as he headed the ball back to Dikgacoi, and the former Fulham midfielder also had the freedom of the Charlton penalty area as he crashed the ball past Hamer from 12 yards.
The hosts, trying to protect an unbeaten home record stretching back to March, blew a fine opportunity to level after 71 minutes when Cort nodded another Wilson free-kick downwards but Danny Hollands scuffed wide from in front of goal.
In four dramatic minutes of stoppage time substitute Ricardo Fuller, recently signed from Stoke, almost made himself an instant hero with the Valley faithful but his deflected effort was tipped on to the bar by Speroni.
And from the corner Garvan cleared a header from Addicks keeper Hamer off the line to secure the points.