With Forest threatening a fightback, Dasilva's outstanding strike made it 2-0 to the Bees in the 83rd minute, before Ivan Toney bagged his 15th of the season five minutes later.
Chris Hughton's Forest were much improved after the break and looked as though they might find a way back into the game before Dasilva and Toney killed the game off.
Joe Worrall pulled one back with a header in injury time and that was not the end of the drama as Anthony Knockaert collected a second yellow card, for simulation, to reduce the home side to 10 men in the 95th minute.
It was Brentford who almost struck first, as Joe Lolley slipped while trying to cross from the right, allowing the visitors to break five-on-two. It ended with a good chance for Toney, who fired wide of the near post.
Brentford did take the lead in the 15th minute, in simple fashion. Mathias Jensen whipped in a precise corner and Dalsgaard, the Brentford captain, sent a simple header beyond Brice Samba.
The Forest keeper produced an outstanding save to ensure Brentford did not stretch their lead, when Bryan Mbeumo let fly with a powerful shot from the edge of the box.
It took 37 minutes for Forest to register an effort on goal but, when it came, Sammy Ameobi forced a decent save from David Raya with a driven shot. Harry Arter's follow-up was easily held by the keeper.
Lolley was not too far away with a curling free-kick as Forest finally found some threat, with Lyle Taylor coming closer still, with a rocket of a shot that fizzed just the wrong side of the upright.
Forest made a double-change at the interval with Knockaert and Miguel Guerrero both introduced and, when Knockaert was fouled on the edge of the box, Raya was well placed to save Taylor's free-kick.
The home side were looking more dangerous and it was a mystery how the ball was not forced home after Ameobi had fired a low cross right across the face of goal, with Knockaert and Taylor both throwing themselves at it.
Forest did have the ball in the back of the net, when Knockaert found the bottom corner. But their celebrations were eventually cut short by a very late offside flag, with Taylor judged to have been blocking the view of Raya.
Forest wanted a penalty after Taylor went down in the box but referee Geoff Eltringham was not convinced and Knockaert's long range effort was tipped over by Raya.
When Sergi Canos fed Dasilva on the break, he produced a spectacular finish to end Forest's hopes of a fightback.
Toney got his customary goal following a moment of hesitation from Samba, with Vitaly Janelt having provided the pass.
Worrall's header, from a Knockaert cross, was no more than a consolation, before Knockaert was then sent off.