The 20-year-old former Arsenal trainee levelled superbly within five minutes of Lee Gregory firing the Lions into the lead but, after Henrik Dalsgaard had seen red for the visitors, neither side could find a winner in a frustrating London derby.
Brentford's defence were too timid as Shane Ferguson drilled a corner into the area and Alex Pearce was able to stab the ball towards Gregory, who struck beyond Luke Daniels for his 12th goal of the season.
But the Bees' response was swift.
Having previously failed to trouble David Martin in the Lions goal, Dasilva received the ball in space 25 yards out and, with time to size up his angles, curled a spectacular left-footed strike into the top corner.
With the fuse well and truly lit, the pendulum swung firmly in Millwall's direction with just over 10 minutes of the first half to go.
A booted clearance upfield set up a foot race between Ben Thompson and Dalsgaard.
The Lions' Thompson nosed ahead in the sprint before he was brought down by the Brentford skipper on the edge of the area, referee Jeremy Simpson immediately charging to the scene to show the Danish international a straight red card.
Once Dalsgaard had departed, Jed Wallace very nearly fired Millwall back ahead, his curling free-kick palmed away low by Daniels.
But on the stroke of half-time, the Bees also came within a whisker of making it 2-1, with wing-back Sergi Canos very nearly recreating Dasilva's earlier firecracker, the Spaniard striking the crossbar with his left boot from an almost identical position.
It was Brentford who looked the more assured outfit in the second half, Dasilva's teasing low ball into the area putting the Lions defence through the wringer.
At the other end, Ferguson saw his chipped cross from the left brush against the woodwork before Brentford substitute Chiedozie Ogbene fluffed the chance to nick it for the visitors, blazing high into the stand after he was picked out by Dasilva, while Bees talisman Neal Maupay rifled an effort narrowly over the Millwall crossbar.
With the spoils shared, Harris' troops are still looking over their shoulders as they battle to stay in the Championship.