The 62-year-old, who led the club from non-league football to League One during his time in charge, replaces Paul Buckle at Kenilworth Road.
Still leaves Dagenham eight points clear of the League Two relegation places, and his assistant Wayne Burnett will take over for the rest of the season.
Dagenham have waived their right to significant compensation, and a club statement said they would instead accept a "nominal" sum "in light of the fantastic job John has done".
The statement went on: "The Daggers board was initially shocked on learning of John's decision but once they realised that it was his wish now to move they quickly decided that to stand in his way would have been wrong given his service to the club.
"The foundations of a great club are clearly there and, as disappointed as many will be, the club would like to formally express its thanks to John for everything he has done and wish him every success in his chosen path."
Still's task now is to restore Luton to the Football League after a four-year absence.
Town are currently 11 points off a Blue Square Bet Premier play-off place, although they have games in hand on all the teams above them.