Making a less auspicious debut on the unfamiliar blue clay, a controversial innovation at this year's edition of the event, was 2010 French Open champion Francesca Schiavone, who was upset 6-4 6-7 6-3 by American qualifier Varvara Lepchenko.
Azarenka, the world number one and runner-up in Madrid last year, dealt efficiently with the potential threat from Russian Kuznetsova, another former French Open champion who had beaten the Belarusian in four of their six previous meetings.
Azarenka has been in blistering form this year, however, with 29 wins, four titles including her first grand slam at the Australian Open which catapulted her to number one, and just two defeats.
Second-ranked Sharapova was coming into the Madrid event following her win over Azarenka in last week's Stuttgart final, her first title of the year and 25th overall, and swatted aside the 56th-ranked Begu who was making her debut at the premier event in the Spanish capital.
The Russian will be looking to complete her collection of grand slams at Roland Garros starting later this month, the only one of the four she has not won.
Former number ones Caroline Wozniacki and Ana Ivanovic had contrasting matches as both progressed into the second round.
Dane Wozniacki, the sixth seed, had to battle past Kazakhstan's Ksenia Pervak 7-6 3-6 6-4, while Ivanovic of Serbia, the 13th seed, dismissed French qualifier Mathilde Johansson 6-4 6-1.
The Williams sisters are in action on Monday, with ninth-seed Serena playing her first-round match against Russia's Elena Vesnina and Venus, playing on a wildcard, meeting Angelique Kerber of Germany in the second round.