The striker hit a 28th-minute penalty too close to young keeper Frank Fielding who saved well. The spot-kick was awarded after Richie Foran had been felled by Leon Johnson.
But Abbott recovered from the disappointment to win the game with a sublime goal on 77 minutes.
He met Joe Colbeck's pass to send a first-time 20-yard shot dipping over Fielding and under the crossbar.
It was tough on the teenage keeper who twice kept out powerful headers from midfielder Michael Cummins who also sent a long-range shot flashing over the bar.
But it was also hard on the entire Wycombe side that were out of luck in front of goal.
Matt Bloomfield hit the post on 15 minutes with a shot on the turn from Gary Holt's cross.
Then as they piled forward in search of an equaliser, Sergio Raul Torres calmly stroked the ball past keeper David Stockdale ten minutes from time only for his shot to strike the foot of the post.
And even after that they didn't give up with the impressive Scott McGleish heading over from a tight angle.
Then Torres had a goalbound shot blocked by a packed defence before substitute Reuben Reid fired over when well-placed.
But over the 90 minutes Darlington shaded it. Fielding was the busier of the two keepers while the Quakers often had Wycombe at full-stretch.
Lively young winger Colbeck and the know-how of Foran and Abbott were in the end too much for them despite the efforts of Fielding and outstanding defender Will Antwi.
Antwi headed away several dangerous crosses from both flanks both before and after Abbott netted the winner.