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)
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