Cirstea rallied to upset fourth-seeded Anastasia Pavlyuchenkova of Russia 4-6, 6-0, 6-4 and Rodina ousted No. 6 Julia Goerges of Germany 6-3, 6-4. Rodionova advanced without playing because No. 9 Flavia Pennetta of Italy pulled out with a left thigh injury.
Rodina, who is ranked 104th, lost in the qualifier but entered the main draw when Germany's Andrea Petkovic pulled out with a knee injury.
Rodina held serve throughout and dominated play against the 21st-ranked Goerges with strong ground strokes.
The 15th-ranked Pavlyuchenkova had 11 double faults and lost eight straight games against Cirstea after winning the opening set. Trailing 2-0 in the decider, she broke back but dropped serve two more times.
In her second quarterfinal of the season, Cirstea will play either Lucie Safarova or Brianti.
Lisicki blew a 5-2 lead on her way to dropping the first set to Brianti. The German rallied to even the match by taking the second set but then failed to take advantage of an early break in the third set as she lost to the 74th-ranked Italian.
Earlier Wednesday, Pennetta was a 7-6 (4), 6-2 winner against Elena Vesnina before withdrawing from her second-round match against Rodionova, who beat Barbora Zahlavova Strycova of the Czech Republic 7-6 (6), 6-1.
Patricia Mayr-Achleitner recovered from losing the opening set to beat Stephanie Foretz Gacon of France 3-6, 6-2, 6-2. The Austrian next takes on top-seeded Wimbledon champion Petra Kvitova for a place in the quarterfinals.
Former champion Daniela Hantuchova missed a match point in the second set tiebreaker but still held off Petra Martic of Croatia 6-3, 6-7 (6), 6-1. Martic had her right thigh muscle taped after winning the second set and seemed to be hampered by the injury in the decider.