Goals from Daggers trio Sam Saunders, Solomon Taiwo and Mark Arber forced Brentford, who would have secured automatic promotion with a win, to put the champagne on ice until the weekend at least, while Still's in-form side moved ominously into seventh place.
Still said: "We've gone for it and that's probably the best performance I've seen in my time here.
"As far as I'm concerned, we have to keep producing with two games to go.
"We've put ourselves in an unbelievable position and it's great for everyone associated with the club.
"I still can't get away from the fact we're Dagenham and to even be in this stage is fantastic, the fairytale continues."