The Daggers had been in terrific form going into the encounter but goals from veteran striker Jamie Cureton and Keith Lowe sandwiched Abu Ogogo's leveller to hand Town three precious points.
Ogogo, Rhys Murphy and Zavon Hines all went close late on for Dagenham but the hosts just could not find an equaliser - with boss Burnett urging his side to learn the lessons from the defeat and swiftly move on.
"I'm really disappointed," he said. "For whatever reason we've contrived to miss too many chances and we've been punished.
"We created plenty of opportunities - particularly late on - but they have to be converted.
"Cheltenham may not have had the best of seasons so far but they've got quality in their side and they're dangerous.
"We have to be better, though. And we will be. We'll have to learn from this result and put it to one side quickly.
"We know we're capable of better, we've proved that over the previous seven games."