Saints made the breakthrough in the 47th minute when Adam Lallana bicycle-kicked Alex Chamberlain's cross past home goalkeeper Tony Roberts before Guly Do Prado calmly side-stepped Scott Doe and buried a shot into the bottom corner in the 76th minute.
Dagenham substitute Josh Scott pulled one back for the home side with a well-placed header and Gavin Tomlin hit the crossbar with a free-kick in the 89th minute as John Still's side piled on the pressure, but Rickie Lambert lobbed Roberts in injury time to seal the points for the visitors.
Adkins said: "As a group I thought everyone contributed. It's not about 11 players, it's about the group and they all stuck together.
"We've come away to a difficult place with great backing from our supporters and great credit to our players, I thought they were magnificent.
"You look at our forwards and you've got Lallana and Chamberlain who were outstanding. We've got (Lee) Barnard coming off the bench and I thought Guly (Do Prado) was outstanding yet again."
Source: PA
Source: PA