Shire started like a team who knew the importance of the fixture and Rodgers and Stephen Brisbane both missed excellent chances in the first ten minutes.
However the former made amends when he raced through on goal and beat Max Currie at his near post.
Colts were well and truly on the ropes and seconds later found themselves 2-0 down when a lapse of concentration at the back allowed Greene to head in unchallenged at the back post from Derek Ure’s corner.
The visitors had Currie to thank for keeping the deficit at two, making an excellent save at point blank range from Paul Sludden’s effort.
Currie’s counterpart Jamie Barclay was called into action for the first time on 20 minutes, when Gavin Lachlan’s free kick from the left after Greene had brought down Stephen O’Neill looked dangerous, but the keeper was able to scramble it behind for a corner after initially fumbling.
Shire’s Jamie McCormack was lucky to escape with just a yellow card after a malicious challenge from behind left O’Neill requiring medical attention, and sparked fury from the Colts players who were clearly growing in frustration as the first half progressed.
Cumbernauld’s Sean Brown came within a fingertip of halving Shire’s lead minutes before the break, meeting a corner from the right with a good header, but Barclay made a terrific save diving to his left to turn it away.
A change of shape for Cumbernauld at half time shifted the dynamic of the match, and the visitors started looking like a more composed outfit.
Despite their confident start, however, the closest they came to testing Barclay in the early stages was a tame effort from O’Neill on the turn from the edge of the box which was easily held by the keeper.
And it was Shire who would strike the next meaningful blow when Richard Kirwan was caught in no man’s land, allowing Rodgers to slot in his second from close range.
Brown looked certain to pull one back for Cumbernauld on 65 minutes when O’Neil’s cross from the left landed on his head, but the striker could only get under the ball and nod it over the bar.
Then Brisbane and Rodgers came close at the other end, the latter with a spectacular long range strike that was well saved by Currie.
But it was Rodgers’ earlier strikes, ultimately, that proved enough to secure the victory for the home side.