Rodgers' ninth-minute opener was quickly cancelled out by Clyde's Scott Ferguson, but the Peterhead striker swiftly restored the visitor's lead.
Midway through the second period Reece Donaldson made sure of the three points with a sweet 66th minute finish.
Stirling Albion spurned a chance to take top spot as they slumped to a 2-1 defeat at the hands of Scottish Cup giantkillers Albion Rovers at Cliftonhill.
First half goals from Barry Russell and Chris Dallas put them in the driving seat.
And Motherwell's conquerors of last week held out for victory despite a 61st-minute header by Jamie Bishop.
Montrose climbed to fifth thanks to a 1-0 win over Queen's Park whose fortunes did not improve at their new temporary home, Airdrie's Excelsior Stadium.
Two goals from Shane Sutherland were enough to seal a 2-0 home triumph for Elgin City against Berwick Rangers.
East Stirling came from behind to salvage a 1-1 draw with Annan Athletic at Ochilview.