Saints' star performer was teenage winger Alex Chamberlain, who celebrated his call-up to the England Under-18s squad with two goals and contributed to another.
The 17-year-old put Saints in front with the aid of Daggers' debutant goalkeeper Chris Lewington, who completely misjudged the teenager's cross as it flew over his head and into the far corner.
But there was more purpose to Chamberlain's second goal when he took aim from a similar position to his first and looped a deliberate effort over the stranded goalkeeper.
Southampton put the game out of reach in the 56th minute when a long kick by goalkeeper Kelvin Davis was flicked on by Rickie Lambert to Adam Lallana, who produced a piece of sublime skill to set up Lee Barnard.
Chamberlain set up Southampton's fourth, showing blazing pace to outstrip Dagenham defender Damien McCrory and standing up a superb cross for defender Jose Fonte to head home at the far post.