Marc White's side downed the National League Ravens with a comprehensive three-goal victory at the Meadowbank Stadium.
Tom Richards, Jack Barham and Jason Prior all netted for the Wanderers as they continued their purple spell.
"I'm over the moon and I thought we were good value for our win," manager Marc White told the club's official video service.
"Bromley are a National League side with 44 points, sitting in the top three - their an established side and a tough opponent.
"But we worked really hard all week and it's been a real team effort to prepare for this - we got a lot of things right.
"I can't wait for the next round," White continued. "To have a big club or an ex-league club down here would be fantastic - we won't fear anybody."
With the second round draw coming Monday lunchtime, the club refocus their attention on league matters.
Ahead of a trip to Maidstone United on December 21, the Wanderers are unbeaten in five in the National League South and they find themselves in the play-off race.
But White feels his biggest task might be to keep his side grounded.
"We're hitting a good run of form and we're doing that whilst managing a few injuries.
"Now we need to really focus on our league campaign because we've had a good run in the league and pulled ourselves into the play-off situation and we want to stay there.
"We're good enough too. But it's a tough old league. We don't want to start thinking we're too good."