After an even first half, Bridges fell to two early second half goals and could have lost by a heavier margin but for a splendid display by keeper Kieron Thorp.
A fierce free-kick by Staf Palmer showed that Molesey weren’t there just for the ride, but Bridges almost stole the lead when a close-range effort by Abu Touray on the turn was somehow headed off the line by Luke Elliott.
Reis Stanislas fired just over for the visitors before Palmer outpaced Matt Daniel to fire in an effort that rattled the crossbar.
But the half ended with Touray forcing Liam Allen to make a good save.
Molesey showed their intent within 30 seconds of the restart with Thorp forced to push away a drive by Dre Grobler, but a minute later Palmer netted off the inside of the post after appearing to control the ball with his hand.
Ollie Robinson then headed just wide from Craig Lewington’s corner, whilst at the other end Touray headed over from Hakeem Adekakun’s pinpoint cross.
But after 51 minutes Palmer doubled his and his team’s tally with another low drive, and he missed a hat-trick chance when he pulled his shot wide a minute after that!
Matt Whitehead fired over following Touray’s deft lay back, but then it was the Kieron Thorp show as he bravely saved at Palmer’s feet having already produced a great save from the same player, while he made two point-blank saves from Stanislaus.
Allen made a good block to deny Adelakun as Bridges refused to give up, and the efforts of both John Lansdale and Ben Godfrey in midfield deserved greater reward.
The very last kick saw another Touray effort cleared off the line, but just before that James McShane was denied by the post and then almost inevitably by Thorp.
Bridges: Thorp, Mobsby, Daniel (Brake 74), Lansdale, Cooney, Crellin, Touray, Godfrey, Tait, Brotherton (Adelakun 46), Whitehead.
Unused subs: Casselman, Sargent, Pulling.
Aerotron Man of the Match: Kieron Thorp