Sam Stosur, Francesca Schiavone, Victoria Azarenka and Elena Dementieva also won but Dinara Safina was beaten.
Venus next faces Italian 15th seed Francesca Schiavone, who crushed Austria's Sybille Bammer 6-2 6-1.
"When you're in a situation like this, every point is like a match point, " said Schiavone.
In the women's section, 16-year-old Ashleigh Barty stunned former French Open champion Francesca Schiavone 6-0 6-3.
The 2010 champion, Francesca Schiavone of Italy, enjoyed a 6-3 6-1 victory over Japanese veteran Kimiko Date-Krumm.
Sam Stosur rounded off Tuesday's play with a 6-4 6-4 win over French Open champion Francesca Schiavone.
Dokic faces Russia's Anastasia Myskina in the next round, who beat Italy's Francesca Schiavone 1-6 6-3 6-4.
Govortsova put out former French Open champion Francesca Schiavone in her opening match at the clay court event.
Italian second seed Francesca Schiavone came from a set down to beat Bulgarian Tsvetana Pironkova 3-6 6-3 6-1.
The Russian was joined in the next round by three former champions: Li Na, Svetlana Kuznetsova and Francesca Schiavone.
She next plays unseeded Francesca Schiavone, the 2010 champion, who beat No. 13 Marion Bartoli of France 6-2, 6-1.
Errani and Francesca Schiavone then combined to beat Hradecka and Kveta Peschke (CZE) 6-2 6-4 in the concluding doubles.
Pennetta's 16th-seeded compatriot Francesca Schiavone was dumped out by veteran American qualifier Jill Craybas, who won 7-6 (9-7) 6-2.
Italy can reach their third final in four years on Sunday when Pennetta faces Kuznetsova and Schiavone meets Chakvetadze.
Mary Pierce doubled France's advantage with a 6-3 6-4 victory against Francesca Schiavone.
Italian 10th seed Francesca Schiavone also booked her place in the next round.
French Open champion Schiavone saved six match points on the way to victory.
Flavia Pennetta and Francesca Schiavone claimed singles wins over Anna Chakvetadze and Svetlana Kuznetsova in Castellaneta Marina in southern Italy.
Schiavone then shocked world number nine Svetlana Kuznetsova 1-6 6-2 6-3 in a match that lasted two hours 21 minutes.
Peer was joined by third seed Yanina Wickmayer of Belgium, and Italian duo Francesca Schiavone and top seed Flavia Pennetta.
Serena Williams put an early break behind her to polish off the first set against Schiavone 6-3 in 34 minutes.
The American will next play Olga Govortsova of Belarus, who upset Italian 14th seed Francesca Schiavone with a 6-4 6-4 victory.
After taking the first three games of the match, Schiavone dropped six of the next seven to lose the first set.
The opening match on Court Philippe Chatrier Tuesday exemplified the trend: Francesca Schiavone, who turns 32 next month, beat 41-year-old Kimiko Date-Krumm.
Despite dropping the first set, Schiavone raced to a 4-1 lead in the second set on her way to levelling the rubber.
Another concept looks too much like a furnace, Mr. Schiavone said, wincing.
"If I can win that match serving like this, that's pretty remarkable, " said the Belorussian, who will next face 2010 champion Francesca Schiavone.
There was a shock at the WTA Aegon Classic in Birmingham as top seed Francesca Schiavone lost 7-5, 6-4 to Japan's Miskai Dothe.
The 22nd seed came from behind to win 3-6 6-3 6-1 against 80th-ranked Romina Oprandi, who eliminated fellow Italian Francesca Schiavone earlier this week.
Rather than relying only on personal inspiration, Mr. Schiavone solicits ideas and proposals from Whirlpool's design studios in Italy, India, China, Brazil and Mexico.
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