A first half of few chances saw Barnet threaten first, though in what amounted to a half-chance at best, top goalscorer John O'Flynn blazed his shot high, wide and not so handsome.
The Bees nearly contributed to the scoring being opened at the other end when, from a Darlington corner, Nicky Deverdics miscued as he attempted to clear. The ball flew across the face of goal with Clayton Fortune unable to convert at the far post.
However, the closest either team came to breaking the deadlock came on the half-hour mark when, from a Deverdics corner, veteran Bees striker Paul Furlong found himself unmarked only to head against the bar.
Albert Adomah then set up Deverdics as the half drew to an end but he couldn't get his shot on target.
The opening moments of the second half presented the clearest opening as Furlong bared down on goal but he was denied by a smart stop from Andy Oakes.
Darlington responded at the other end when the returning Liam Hatch somehow failed to convert from a right-wing cross, although they didn't have to wait long to find the back of the net.
Danny Carlton did well on the touchline before laying the ball to Abbott who slotted the ball home via a Barnet defender's attempted clearance.
Darlington lost Oakes for the last ten minutes although his replacement, Przemyslaw Kazimierczak, had little to do, tipping over Joe Devera's shot and brilliantly clawing away Yannick Bolasie's shot.
Ismail Yakubu came closest to salvaging a point though when his header from Deverdics' corner was cleared off the line.