Other seeded players to progress were China's Na Li (16), Daniela Hantuchova (22) and Shahar Peer (29).
Fourth seed Venus followed up her opening win over Patty Schnyder with another routine straight-sets victory, defeating Israeli 16th seed Shahar Peer 6-3 6-4.
Meanwhile, Russian second seed Maria Sharapova was given a scare, dropping the opening set before rallying to defeat Israel's Shahar Peer 4-6 6-3 6-3.
German 14th seed Sabine Lisicki will play Israeli Shahar Peer in round two after a hard-fought 6-2 4-6 6-4 win over Switzerland's Stefaine Vogele.
Williams will now play Shahar Peer in round three after the Israeli recovered from dropping the opening set to beat Polona Hercog 2-6 7-5 6-3.
Clijsters, the 14th seed but champion in Miami in 2005, crushed 20-year-old Czech Petra Kvitova 6-1 6-1 to earn a third-round clash with Israel's Shahar Peer.
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Meanwhile, in the women's draw, Kim Clijsters -- who is only seeded 14th for the event -- crushed 17th-seed Shahar Peer 6-0 6-1 in just 51 minutes.
Top seed Caroline Wozniacki defeated Slovakia's Dominika Cibulkova 6-2 7-6 (7-2) and the Danish world No. 3 will play Israel's Shahar Peer for a place in the quarterfinals.
Russian second seed Anna Chakvetadze beat Israeli Shahar Peer 6-3 6-4 and plays the winner of Thursday's second-round match between French sixth seed Marion Bartoli and Britain's Anne Keothavong.
Ninth seed Flavia Pennetta also went through with a 6-3 6-3 win over Czech Petra Kvitova, and the Italian will play Israel's 17th seed Shahar Peer in the third round.
Earlier on Friday, defending champion Venus Williams cruised into the final of the Dubai Open as she ended the surprise run of Shahar Peer with a 6-1 6-4 win in the semifinals.
Tenth seed Shahar Peer completed a routine 6-3 6-2 victory over Romania's Sorana Cirstea to secure a match against Italy's 22nd seed Flavia Pennetta , who beat Spain's Lourdes Dominguez Lino 6-2 6-2.
Ivanovic had been distraught after a first-round defeat by Shahar Peer in San Diego last week and she looked set for another early exit when Azarenka served for the match on Monday night.
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