Bradshaw pulled off a series of stunning saves to help his side do the double over their Sussex Hockey League Division Two opponents having won the home fixture in November.
Match Report from Sol Buckner
Striker Ben Simmons made Saxons' intentions known in the first minute when his clever lob over the oncoming keeper from just inside the D cannoned off the far post.
Then, minutes later, stand-in skipper Paul Merison's snapshot from a short corner was touched in to the goal by a defender.
Saxons then controlled the first half, limiting Crawley's chances and taking a deserved lead into the break.
The second half saw the visitors continue to take the game to their hosts as they searched for a second goal.
And it arrived in the 40th minute when Sol Buckner latched onto a ball at the top of the D, slipped past a defender and fired past the outstretched goalkeeper.
Saxons' defence held tight marshalled by the brilliant Bradshaw who continued to frustrate the hosts.
A quick break in the 47th minute resulted in a short corner which Dom Whitaker converted well with the help of a defender's stick.
Crawley pushed hard in the final quarter, but despite half-a-dozen short corners, could find no way past a resolute defence.
Talented youngster James Chu earned man of the match.
South Saxons II welcome Lewes IV to Horntye Park this Saturday for a 12pm push back.