The 30-year-old, a wildcard entry for this week's ATP event, broke his opponent three times before sealing victory in a little under an hour and a quarter.
Back in action last month after being sidelined for a year following two surgeries on his left wrist, Kiefer kept the Russian under continual pressure with deep, probing groundstrokes.
He won the opening set after breaking Kunitsyn in the eighth game and powered through the second after further breaks of serve in the first and fifth games.
Although the Russian saved two match points, one on his own serve and the other after one of the best rallies of the match, Kiefer retained overall control in searing heat at the Los Angeles Tennis Center.
After surprisingly trailing 0-30 on his own serve in the eighth game of the final set, the German unleashed a series of big first serves to claw his way back and clinch the win.
Kiefer, playing in his first hardcourt tournament in 16 months, set up an all-German quarter-final clash with compatriot Michael Berrer.
Berrer, who broke into the world's top 100 just before Wimbledon last month after reaching his first ATP quarter-final at 's-Hertogenbosch, beat Brazilian Ricardo Mello 6-3, 7-5 in an earlier match on Wednesday.
Seventh-seeded Korean Lee Hyung-taik also reached the last eight after his South African opponent Wesley Moodie retired with a back injury when trailing 6-2, 3-0.
Former world number one Marat Safin was scheduled to play Swiss George Bastl in the day's late match on the stadium court.
