The Hornets had a hatful of chances in the first half but a strike two minutes before the break from Darren McQueen gave the Darts the lead.
The second half followed the same pattern as Horsham bombarded the Dartford goal but were kept out by keeper Mark Smith.
The Hornets' FA Cup exit was confirmed when Luke Allen fired home at the death.
Manager Dominic Di Paola made two changes from Tuesday night's 2-1 home win over Wingate & Finchley. Goalkeeper Josh James made his first start following his return to the Hornets in the summer. Kieran Lavery returned from injury and replaced Kieron Pamment.
On three minutes, a ball over the top from Ryan Hayes tried to pick out McQueen but James was quick off his line to gather the danger.
Three minutes later Horsham had their first sight of goal. Charlie Harris let fly from outside of the box but his long-range effort went wide of the post.
Patient build up play from the Hornets on 11 minutes saw Harvey Sparks play a delicious outside-of-the-boot-pass into Chris Smith on the edge of the area. The forward unleashed a powerful strike that Darts keeper Smith could only parry out.
A minute later a Smith snapshot looked destined for the bottom corner but Smith did well to turn the ball away for a corner. Harris' corner found the boot of Joe Shelley but the defender rattled the post.
The Hornets should have taken the lead on 18 minutes after having four chances in a matter of seconds. A lovely flowing Horsham move saw the ball find Harris in the box. The midfielder had multiple chances blocked as Dartford defenders put their bodies on the line to keep him out. The ball fell to Lee Harding who struck hard and low at goal but a terrific goal-line clearance from Josh Hill denied the winger.
The Hornets were controlling the first-half and had another super chance to go in front on 27 minutes. Harris' whipped free-kick found Lea Dawson completely unmarked in the centre of the area. But the midfielder's close-range header was magnificently tipped away by Smith.
But the Darts took the lead against the run of play three minutes before half-time. A deep cross from the right by Hayes was headed across goal by Hill. McQueen met it and produced an acrobatic finish to put the Kent side a goal to the good.
11 minutes into the second half, an inch-perfect tackle prevented Horsham netting the equaliser. Good link-up play between Smith and Harris saw the former pull the ball back to the latter in the box. Harris looked odds-on to score but a marvellous sliding tackle saw the ball taken away from the midfielder.
The Hornets had two more gilt-edged chances to get themselves back on level terms. On the hour a driving run from substitute Zack Newton saw him pick out Sparks in the area. The left-back steadied himself and unleashed a ferocious shot that stung Smith's fingertips.
Four minutes later Steve Metcalf fired a cross-cum-shot into Shelley but his flicked attempt was skewed wide.
Dartford hit back with a chance of their own on 69 minutes. A Ben Greenhalgh free-kick met Jordan Wynter at the front-post but his looped header went millimetres wide.
With 78 minutes to go, Smith was unlucky not to level. The striker received the ball at close-range and did well to get off a shot after struggling to get the ball out of his feet.
With eight minutes to go, Kieron Pamment broke through and drew another top-draw save out of Smith - although the offside flag was raised.
And to compound Horsham's luckless day, Allen struck at the death to make it 2-0 Darts.
Horsham: James, Metcalf, Sparks, Kelly, Shelley, Miles (Merchant 73), Lavery (Newton 60), Dawson (Pamment 78), Smith, Harris, Harding. Unused: Bentley, Mills, Brivio.
Referee: Shelby Elson