Hartley nodded home his first Boro goal inside the first 10 minutes before Doughty notched late on to hand Graham Westley's side a fourth win in five matches in all competitions.
Crawley began brightly and debutant Kyle Bennett saw his 25-yard screamer parried wide by Chris Day in the first minute.
But the home side took an early lead - Hartley sending a powerful header past Paul Jones from Greg Tansey's eighth-minute corner.
Nicky Adams almost finished a fine Crawley move after 40 minutes but his low 20-yard drive flew just wide while Dannie Bulman and Josh Simpson both missed good chances before the break.
The Red Devils started the second half where they left off and only Jon Ashton's goal-line clearance prevented Adams equalising after 50 minutes.
Day's heroics denied Crawley's James Hurst a leveller in the dying minutes before Doughty sealed the points in injury time, cutting off the left flank to fire an angled shot into the far corner past Jones.