Teenage full-back John Flanagan provided a testing cross from the right which Mike Williamson only partially headed clear, allowing Rodriguez to volley a shot from 12 yards that took a crucial deflection off Danny Simpson past keeper Tim Krul.
Myler's second successful conversion put Northampton 20-14 in front at the break and Gloucester reshuffled their back division for the second half, with Simpson-Daniel moving into midfield instead of Tim Molenaar and Charlie Sharples slotting in on the wing.
The others are Leicester duo Anthony Allen and George Chuter, Wasps pair Joe Worsley and Tim Payne, Gloucester's James Simpson-Daniel and Paul Doran-Jones, Saracens wing David Strettle and Harlequins flanker Chris Robshaw.