In the men's tournament, Novak Djokovic withdrew before the start of the event because of a right wrist injury that had already affected his play at Monte Carlo last month.
"I did everything possible in order to play in Madrid," Djokovic said. "But unfortunately my right arm injury has flared up again."
The top-ranked Williams, who has won the clay-court tournament the past two years, hit 11 aces and broke Bencic five times to advance in less than an hour.
Williams will play Peng Shuai of China in the second round.
Sharapova, who lost to Williams in last year's final, converted six of eight break points to take advantage of Koukalova's eight double-faults.
The eighth-seeded Sharapova will play Christina McHale of the United States next.
Also, Ana Ivanovic beat Madison Keys of the United States 6-1, 7-6 (4) to set up a meeting with fellow Serb Bojana Jovanovski. The 11th-seeded Ivanovic overcame her shaky serve by saving three of four break points.