The Most Valuable Player of the Reds' World Series championship team in 1990, Rijo has battled arm problems through much of his career and underwent five surgeries before rejoining Cincinnati as reliever last season.
Rijo (3-1), who now banks on location rather than velocity to get hitters out, limited the Brewers to three hits with four walks and a strikeout.
Jim Brower, Scott Sullivan and Luis Pineda combined to allow two hits over the final four innings.
Kearns homered off starter Jose Cabrera (1-4) in the fifth to extend the Reds' lead to 4-0.
Making his first major league start after 167 relief
appearances, Cabrera surrendered four runs and six hits in 4 1/3 innings.
Cabrera got the call in place of Nick Neugebauer, who was scratched before the game with discomfort in his right shoulder.
Notable deaths of 2014 [PHOTOS]