An opening day win over Barrow was followed up at Wood, with the Daggers aiming to push on after last season's play-off defeat - watch highlights from last night's game here.
An early goal from the visitors' Fejiri Okenabirhie after just three minutes was quickly cancelled out by Bruno Andrade's penalty moments later.
But Dagenham retook the lead in the 19th minute through Corey Whitely's stunning free-kick, and that proved to be the decisive goal.
"Local derbies, I think, are the hardest games to win," Still told the club's website. "I thought we played very well first-half and we didn't play as well in the second half but what we did do was battle.
"A local derby, away from home, I'm delighted to have won. Can we play better? Yes. We've played two games and we've shown what we're capable of in both.
"It's a long season, I never get too excited. I think it takes 6 to 8 games for everything to sort itself out so whatever we can get in these games is going to be a big plus for us."