Manager Mark Bower said: “It was an important win for us and this was our most complete performance since we gained promotion. From one to 11, we played really well and the first goal was fantastic.”
Guiseley’s Will Hatfield soon broke through in the windy conditions, which prevailed throughout, but his pass across the face of goal found no one.
Moses Emmanuel’s direct running then started to trouble the Guiseley defence before they responded and Norburn’s deflected effort hit a post.
With first-half stoppage-time looming, Norburn again got the ball 30 yards out and hit a tremendous effort which flew into the top corner.
The visitors started the second half well with Reece Prestedge’s free header hitting the top of the bar.
After withstanding that early pressure, Guiseley should have got a second on several occasions but Bromley goalkeeper Chris Kettings made five great saves in the space of four minutes, the best one coming when he tipped over a shot from Danny Lowe.
But Dudley was able to seal the win late on with a tap-in after Kettings had made another good save.
This victory sees Guiseley end a run of three loses and prepares them for the local derby away to next-to-bottom FC Halifax Town on Saturday.
Guiseley: Drench, Hurst, Lowe, Lockwood, Lawlor, Smith, (N.Boshell 85) Norburn, Hatfield, Boyes (Craddock 90) , Dudley, Rothery. Unused substitutes: Toulson, L Dickinson and Atkinson.
Bromley: Kettings, Udoji (Wall 62), Turnwa, (Connor 75) Fuseini, Swaine, Holland, Joseph-Dupois, Prestedge (Ajakaiye 75) Minshull, Emmanuel and Chorley. Unused substitutes: Rogers and Dewer-Douglas.
Referee: R Wigglesworth (Rotherham).
Man of the match: Chris Kettings.