In a rather dull first half at The Crabble it was the home side who had the first chance of the game.
Former Bromley forward Moses Emmanuel controlled the ball well in the box before beating his man and getting a left foot strike away but a Flitney was able to dive low to his left and tipped the ball onto the post before the danger was cleared.
Bromley’s best chance of the half came when a counter attack led by Dennis down the left saw the winger cut into the box before pulling the ball back to Sho-Silva, his effort was blocked by the defender before another defender managed to beat Goldberg to the ball before clearing the danger.
Flitney was called into action early in the second half when he had to save a powerful effort from Lafayette inside the box.
The home side thought they had a good shout for a penalty minutes later. Modeste went down inside the box under pressure from Anderson but nothing was given.
Dover took the lead on the hour mark when a good cross into the near post was palmed away by Flitney but only into the path of Healy before trickling over the goal line.
Bromley looked to respond, Goldberg was put through by Minshull but his effort, under pressure from the recovering defender, was straight at the keeper.
Former Bromley striker Moses Emmanuel continued to look dangerous for the home side as they looked for their second, a curling effort from Emmanuel sailing over the bar minutes later.
The best chance for an equaliser came with five minutes to go, Dunne heading a corner at goal only to see it headed off the line and over the bar.
As the full time whistle went it was all three points for Dover however they appeared to end up a man down following an altercation after the whistle where it looked like the referee issued a red card.