The visitors stormed into a commanding early lead as John-Joe O'Toole, Oumare Tounkara and Danny Woodards all found the net, before the Daggers pulled a goal back before half-time through Ben Strevens.
Tounkara registered his second goal in the 70th minute, with substitute Jake Reed tapping home Femi Ilesanmi's cross to make it 4-2 but it proved mere consolation.
After presiding over a fifth match without a win, Burnett said: "Once you concede three goals inside the opening 20 minutes in a game you're always going to be up against it.
"That's certainly been the case. We've lost concentration too many times at the back and we've ended up giving ourselves a massive mountain to climb.
"Fair play to Bristol, they've been clinical up front, but we can't be defending in that fashion.
"If we're not careful we're going to be getting ourselves dragged into trouble. We're a few games without a win now and that's a concern.
"The players are down now and that's not surprising, but we need to get back to the kind of play we know we are capable of producing.
"A bad home defeat doesn't make you a bad side overnight. We've got to move on from this very quickly."