Bridewell took the lead off Bradley Ray on lap six, but Redding came through a few corners later and was able to maintain his lead.
Brookes took advantage of a lacklustre exit from Bridewell, sneaking ahead with Ray three seconds further back in fourth.
It was quite the finale with the Be Wiser Ducati rider, Brookes and Bridewell all jostling in the final lap.
Redding held on with some high class defensive riding, while his team-mate Brookes managed to steal second from Bridewell.
Christian Iddon kept his Showdown hopes alive as he moves eight points behind Peter Hickman, who occupies the final top six spot.
Tarran Mackenzie also moved closer to securing his BSB Showdown place with ninth for McAMS Yamaha, going 20 points clear of his nearest rivals outside the top six.