The Railwaymen defended soundly but lacked the killer punch in front of goal and rarely threatened the visitor's defence.
Dagenham, with little to play for but pride, also struggled to create chances but one goal on the half hour proved decisive.
Leigh's first clear-cut chance came in the 10th minute when Lee Mulvaney's header put Ben Jones in the clear but he fired way off target.
Michael Simpkins rescued Leigh when he took the ball off the toes of Craig Mackail-Smith as he raced into the box, but with their first real chance of the game on 30 minutes Dagenham scored.
Paul Bruce's corner was flicked on by Jake Leberl and Chris Moore made no mistake with a header from just one yard out.
Mackail-Smith was the visitor's most dangerous attacker and twice he forced Jon Worsnop into making saves.
In the second half Leigh toiled hard to little effect but they could have levelled with seven minutes remaining when Craig Smith's cross found the head of substitute Leam Coyne but he could not direct it on target.