There was a nervy moment for Dagenham goalkeeper Chris Lewington in the early stages, as he slipped on the edge of the box while attempting a punt forward, but his defenders cleared to avert the danger.
At the other end, Stanley keeper Marcus Bettinelli was looking more composed and he saved well from Hines, as well as comfortably holding a couple of Daggers free kicks.
However, he was beaten twice in quick succession after the interval, with Hines netting the opening goal when he combined with Murphy before poking past the Fulham loanee on 56 minutes.
Roles were reversed five minutes later when Hines turned provider, releasing Murphy to send Bettinelli the wrong way with a composed finish to a swift Dagenham counter-attack.
Accrington player-manager James Beattie named himself among the home substitutes, but did not bring himself off the bench to try to inspire a Stanley comeback.
Although Gray bundled the ball home from close range on 91 minutes following a spill from Lewington, it was too little too late for Beattie's side, who are still winless and bottom of League Two.