Goals moments each side of half-time cost Bridges further FA Cup glory in a 4-1 defeat, but they started brightly before Bostik League high-flyers Greenwich Borough were able to stamp their authority on the game thanks to a wealth of experience within their ranks.
Lee Hall fired just over in the first three minutes before Bridges took a seventh minute lead as Nathan Crabb played the ball inside to Connor French, who turned delightfully before firing past Ben Bridle-Card into the bottom corner of the net.
But within another seven minutes Borough were level when a corner reached Francis Babolala, who had time to control the ball before firing across the goal into the far corner.
But the rest of the first half saw both sides cancelling each other out and it seemed to be meandering to a stalemate until Liam Collins was forced to knock a ball towards the corner flag.
Bradley Peters, who had a fairly impressive full debut for the side, was put under pressure by the dangerous Jack Sammoutis, who worked his way along the goal-line before slipping the ball into the path of former Crawley Town favourite Charlie MacDonald for a simple tap-in.
That came just two minutes before the interval, but within a minute of the second half getting under way, Bridges fell further behind when MacDonald turned the ball back neatly for Travis Gregory to guide the ball past an unsighted Kieron Thorp.
Injuries forced Bridges into a swift double substitution and Thorp did brilliantly to keep out another effort from Gregory as a now relaxed home side began to dominate proceedings with Bridges often reduced to hopeful long balls to the obvious frustration of Crabb and French in particular.
The ever combative Jamie Crellin had a late effort which Bridle-Card gathered, and with just two minutes remaining Bridges rather cruelly conceded again when substitute Connor Hunte cut in from the right and hit a good shot past a helpless Thorp.
Bridges: Thorp, Collins, Newman-Knott, Crellin, Conroy (Lansdale 52), Peters, Crabb, Grant (Odofin 52), French, Hall, Johnson (Hilaire 64).
Unused subs: Casselman, Lombardo.
Aerotron Man of the Match:
This Saturday Bridges are at home to Eastbourne Town in the league, while on Tuesday, August 29 they revisit Horsham YMCA, also in the league.