After a goalless first half - and wholesale changes for the Clarets at the break - Rodriguez opened the scoring in the 56th minute at Boundary Park.
The striker turned the ball home from close range after defender Bobby Thomas guided Dwight McNeil's corner goalwards.
Sean Dyche's side extended their advantage a minute later when Richardson pounced on a loose ball to beat the Latics' trialist goalkeeper.
First half: Peacock-Farrell, Bardsley, Tarkowski, Mee, Pieters, Gudmundsson, Cork, Benson, Glennon, Barnes, Richardson.
Second half: Norris, Lowton, Collins, Thomas, Taylor, Gudmundsson (Gomez), Westwood, Brownhill, McNeil, Rodriguez, Richardson (Thompson).
Attendence: 2,071 (982)