The 37th-ranked Wickmayer recorded her third career top-10 victory with the win over 10th-ranked Bartoli, and the first of the season.
Vandeweghe, ranked 120th, earned a spot in the main draw after three players withdrew because of injuries.
Williams, the only higher seed to win on the day, said she's still affected by jet lag and did not get much sleep Thursday night.
"I woke up at 1 a.m. and didn't want to take anything so I just stayed up," Williams said. "At least I got to finish 'Desperate Housewives,' my favorite series ever. I was crying as always. It's so sad it had to end."
Williams started slow and finished strong.
"I knew I was going to start slow, but I expected to stay consistent and hang in there," she said. "Finally, I started moving better."
Williams and Cirstea will meet for the first time, and it's a match that Williams has been looking forward to play.
"I think it's going to be a good match," Williams said. "She has a good serve, which is why I love her game."
Cirstea, who will represent Romania at the Olympics, improved to 5-7 against top 20 players with her win over Cibulkova, who won their two previous meetings.
"I did not feel comfortable at the start," Cirstea said. "Then I got used to her pace and I served pretty well. I was playing better match to match."