Second seed Svetlana Kuznetsova also won, but produced an erratic performance in defeating the tenacious Anna Smashnova 6-2, 4-6, 6-0 in her first clay court match of the year.
The seventh-seeded Clijsters looked far more comfortable than she had in her opening round struggle with her strength of shot often forcing her 18-year-old opponent into errors.
Kirilenko held serve just once in the first set, and even then faced a break point, and she held only once in the second as Clijsters dominated.
Clijsters, playing her fourth tournament of the year after claiming victory at Indian Wells and Miami, dropped her serve after breaking to lead 2-0, but was never troubled by a promising but lightweight opponent.
After taking the first set, Clijsters began the second in style, claiming the first three games for the loss of just one point.
Looking increasingly at ease, Clijsters fired a fierce return to break again for a 5-1 lead, before closing out the one-sided contest
"I was seeing the ball a little better today and didn't make all those unforced errors," said Clijsters. "In my mind as well I was positive, focused, and it was definitely a big step forward compared to yesterday. My serve was on most of the time. I felt good and was hitting the lines."
U.S. Open champion Kuznetsova began strongly against an aggressive opponent who retrieved well but hit few winners. Three breaks gave the Russian the opening set, and she then held three points for a 3-0 lead in the second.
But she was unable to maintain her momentum and a succession of hurried errors allowed the Israeli to level at one set all. The final set, however, saw Kuznetsova regain control as she gave up just two points on serve.
"I didn't play for three weeks and it's my first match on clay, and it was a little bit tough," said Kuznetsova.
"I had to play longer points and I wasn't ready for that mentally."
In other matches, fifth-seeded Russian Elena Bovina edged through 6-4, 4-6, 6-3 against Slovak Daniela Hantuchova while Italian Silvia Farina Elia, seeded eight, overcame Czech qualifier Zuzana Ondraskova 6-1, 6-2.