The visitors started brightly as Harry Forrester floated the ball just over the bar when taking aim from 35 yards.
Douglas should have done better when he met Ryan Fredericks' far-post free-kick before Owain Fon Williams pulled off a wonderful one-handed save to divert Forrester's neatly-worked effort for a corner eight minutes before the break.
Tranmere almost took the lead after 57 minutes when Jean-Louis Akpa Akpro flashed a header just past the Brentford upright from James Wallace's cross. However, he made amends by beating Shaleum Logan all ends up and cutting in to roll the ball into the net just before the hour.
Ash Taylor nearly doubled Rovers' advantage when he tried to steer a ferocious Liam Palmer strike towards goal soon after, while Jake Cassidy forced Bees 'keeper Moore to react when he took two ultimately fruitless bites at the cherry.
Akpa Akpro then saw another chance go close, but Rovers' defence was then guilty of letting a long throw bounce inside their penalty area and Douglas volleyed home an equaliser deep into injury time.