The No.3-seeded Azarenka was the first to advance, though it probably wasn't the way she wanted - she was leading Ayumi Morita, 61 20, when the Japanese No.1 had to retire with a left thigh injury.
"Your opponent getting hurt is never the best way to win your matches," Azarenka said. "I hope it's nothing serious and that she can continue playing again very soon, maybe next week or something."
Next up for Azarenka is No.9 seed Samantha Stosur, who pulled off a 75 26 61 upset over No.8 seed Petra Kvitova. Though Stosur was only seeded one spot below Kvitova, it was still a big surprise - Kvitova had won all four previous meetings, including a 62 61 rout in their only clay court meeting.
Stosur Vs Kvitova Video: Stosur Interview
Azarenka has beaten Stosur in all seven of their previous meetings, but their last one was by far the closest - after losing their first six in straight sets, Stosur pushed Azarenka to a third set tie-break in the quarterfinals of the US Open last year, just two points from victory at 5-all in the breaker.
"She's a very high quality player and a Grand Slam champion," Azarenka said of Stosur. "She's always a very difficult opponent for me but I'm looking forward to it. I'll have to play my best against her."
At night, the No.2-seeded Sharapova overpowered No.16 seed Sloane Stephens in straight sets, 62 61, to continue her path towards her third straight title here - she won back-to-back in 2011 and 2012.
"I expected a tough match from Sloane - obviously she had a great start to the year, getting to the semifinals of the Australian Open, so I know she's capable of playing really good tennis," Sharapova said. "It was important for me to start well today and finish the match off on a good note too."
"I played a good match in singles and we won in doubles too, so it's been a really good day," Errani said. "In the singles we played a very nice first set, then she unfortunately had to retire because of her knee. But I'm very happy to be in the quarterfinals here in Rome. It's a very special feeling for me."
By virtue of the day's results, Errani will make her Top 5 debut on the next rankings, moving from her current ranking of No.6 to a new career-high of No.5. She will be the second Italian ever to reach the Top 5, after Francesca Schiavone, who made it as high as No.4 in the world during the 2011 season.
Errani Vs Kirilenko Video: Errani Interview
Sharapova has beaten Errani in all four of their previous meetings, but while the first two were routine, the last two - at Indian Wells and Miami earlier this year - were some very close two-setters.
"She's probably going to be tougher to play at home, with the Italian crowd helping her," Sharapova said. "We have had some tough matches and I've won them, but she is an extremely good player."