Despite the scoreline, this was a much improved display by Bridges against a side packed with experienced former pros, including a couple previously with Crawley Town.
But after the debacle against Chipstead, Bridges more than matched their visitors for much of the game, suffering only in a 15-minute spell in the second half when three goals were conceded.
Abu Touray showed early intent for Bridges with a shot just over the bar, but Greenwich responded with the impressive Harry Donovan firing just wide after good work by Malachi Hudson.
A cracking volley by the non stop Lee Hall rattled the crossbar as keeper Craig Holloway just got his fingertips to the ball, but Bridges finally went ahead after nineteen minutes through a penalty.
John Lansdale was pulled to the ground as Alex Rose’s corner came over and Tony Garrod drove home the spot-kick.
It took just three minutes for Greenwich to respond as Donovan drove through and slipped the ball into the path of ex-Crawley favourite Charlie MacDonald to equalise.
The rest of the first half continued to be end-to-end with Bridges possibly even looking the more dangerous side.
But Liam Hickley was proving to be a stubborn rock at the back for an otherwise sometimes shaky visiting backline.
Lee Hall came close with a header from Rose’s free-kick as the second half got under way and Bradley Pritchard saw a solid effort deflected away before Greenwich took a 58th minute lead through a snap shot by Travis Gregory, this being the 100th League goal conceded by Bridges this season!
A reoccurrence of a hamstring injury ended an impressive performance by Adam Thompson at the back for Bridges, although his replacement Oneil Odofin - who had been delayed because of a cancelled train - looked equally cool.
But the cruelty of football when you’re down at the bottom was shown when a high ball near the half way line cannoned off Rose’s back and left him off balance as Hudson collected the loose ball and scampered away down the right.
His shot ballooned high in the goalmouth and the last touch was credited to Gregory almost on the goalline itself.
That goal came after 67 minutes and six minutes later MacDonald threaded a delightful ball through for substitute Mohamed Eisa to score with his first touch.
Greenwich, who looked destined for the play-offs, threatened a few more times in the closing stages and Martin Grant was forced to a brilliant tip over to deny Lewis Spence.
But Bridges had moments of their own too and certainly didn’t deserve to lose by such a margin.
Bridges: M.Grant, Lansdale, Rose, Hall, Thompson (Odofin 64), Sargent, Touray, J.Grant, Garrod (Pierre, 80), Cook, Fender.
Unused subs: Allen-Djilali, Adelakun, Mobsby.
Aerotron Man of the Match: John Lansdale