Dwight Gayle pounced after Michael Spillane's spot-kick had been saved to put the Daggers in front at the break, but former Dagenham full-back Damien McCrory and substitute Matt Paterson both scored with the space of three minutes to give the home team the lead shortly after the restart.
Substitute Sam Williams smashed home to level the scores in the 78th minute, but it took Burton only a minute to regain the lead thanks to an equally good finish from the edge of the area by Billy Kee.
Veteran manager Still said: "I thought it was a careless and naive performance from us. We got the goal but then in the space of two minutes we let them get in front when we should have done a lot better defensively.
"Their third goal was a good strike but again it stemmed from defensive naivety. Our two centre-halves just weren't good enough because two goals away from home should be enough to get us something from the game.
"We've been tremendous of late and we've had a great run, which makes this result all the more disappointing and frustrating. I think we're a team getting back to where we want to be in terms of attacking and scoring goals, but we have got to defend better than we did in this game."