The wining goal came after 19 minutes when Matt Saunders crossed from the right for Sam Williams to send a right-foot volley past home goalkeeper Sam Walker from 12 yards out.
The Daggers had been kept on level terms by an amazing triple save by Chris Lewington after 15 minutes as he dived to deny Fabian Broghammer and David Clarkson twice.
Still said: "Whenever we beat a club the size of Bristol Rovers it is a great day for us.
"I am used to having to sell my best players and work with one of the lowest wage bills in the league. But we have a team of fighters and that showed.
"The triple save was a big moment in the game and we then scored a very well-taken goal.
"From then on we had to dig in. My back-four had an average age of around 22 so I am very proud of them."