Simpson, who will remain available as a player, took temporary charge following the departure of long-serving Rod Stringer in late January.
The Clarets are unbeaten in four and have yet to be defeated at home since the caretaker striker was given the reins seven weeks ago.
Having helped with the creation of the Club's Academy at the start of the season, Simpson hopes to maintain the pathway of the Club's young players into the senior squad.
Robbie Simpson said; “I feel very proud to be appointed as the new manager.
"I think this Club has fantastic potential and I am really looking forward to playing my part in the next phase of the journey towards being a truly successful ‘community' Club.
"I’m very passionate about the Academy and I want to continue giving youngsters a chance."
Sitting tenth in Vanarama National League South, Chelmsford City are six points away from the play-off places with a game in hand.
Photo Credit: Matt Bradshaw