Despite being under the cosh for long periods against promotion hopefuls Dale, Rhys Murphy's 63rd-minute goal proved enough for the Daggers to earn their first win on the road since October.
"It was a little bit of a smash-and-grab raid, but I'm sensing a little bit of disappointment from the supporters," said Hill.
"Yes, we are disappointed to lose but let's have a reality check and a sense of honesty here - we are going to lose games, no question about that. I cannot spend two or three million pounds, I've got honest players and we lost honestly.
"We should have made more of the possession we had in the first half. You cannot allow the opposition to grow into the game. We knew they would set up to counter attack and we know they are very good at counter attacking. We looked an opportunity in the eye and really threw it away.
"We've been beaten by a well-organised side who can hurt you on the counter attack. The first goal was always going to be key and we had opportunities to get it but we didn't."
The home side wasted a great early chance to score when Ashley Eastham's deflected header was blocked by Dagenham goalkeeper Chris Lewington. George Donnelly was left to apply the close-range finish from the rebound, but the striker blazed over the crossbar from four yards.
Lewington pulled off an acrobatic save to keep out Donnelly's header and later blocked Matt Done's effort from a tight angle as Rochdale dominated the contest.
But Burnett's side showed more ambition at the start of the second half and took the lead when Luke Wilkinson's delivery from the left was turned in by Murphy.
Scott Hogan had a header cleared off the line while Lewington saved another piledriver from Done to frustrate the home side's attempts to force an equaliser.
Dagenham should have doubled their lead when Medy Elito found substitute Chris Dickson six yards from goal only for Dale stopper Josh Lillis to save.
- Sports & Recreation
- Rhys Murphy
- Chris Lewington