On 83 minutes, Matt Lench delivered a corner which was met by George Wells in the crowded penalty area, the ball evading Charlie Grainger who was unable to prevent it crossing the line.
The vociferous Rebels celebrated jubilantly behind the goal.
The comeback begun midway through the second half with a leveller courtesy of the feisty Harris.
Lench delivered a driven corner to the far post where Harris powerfully headed back across goal into the far corner.
Prior to this, the Rebels rarely threatened but Harris did come close.
Scott Davies launched a high ball into the box, which was meat by the leaping Harris, whose header was tipped over the bar by the Dulwich stopper, who had a relatively quiet afternoon until that point.
On 38, the hosts deservedly opened the scoring after a short corner was well worked to Clarke, who had enough space to bend a ball beyond the scrambling Jonathan North.
Slough had a half chance of their own just before Hamlet scored. Harris created some space for himself before, swiveling in the box before striking his effort which was comfortably held by Charlie Grainger.
The Rebels had the game’s first opportunity - Johnny Goddard scuffed his effort from outside the area, when he ought to have tested the keeper.
The hosts managed to get a foothold and assert some authority with Nana Owusu curling an effort from the edge of the box just wide.
Dulwich created more chances, and on 17 should have opened the scoring.
Clarke was found with time and space inside the Rebels area, but he side footed his effort wide with the goal gaping.
Soon after Clarke failed to connect from a cross down the right with the goal gaping, North was left stranded as the ball trickled past the post.
After being on the back foot for much of the first half, Slough composed themselves and managed to grind out a hard-fought, morale-boosting victory.
Image: George Beck
Dulwich Hamlet Line up
Charlie Grainger, Jack Holland (c), Frankie Raymond, Danny Mills (Michael Timlin 87), Sam Deadfield, Nana Owusu, Quade Taylor, Andre Blackman (Kreshnic Krasniqi 88), Jayden Clarke, Joe Felix (Dan Smith 76), Ronnie Vint. Unused substitutes: Jamie Splatt, Junior Savane.