Williams, who is looking for her third title at Roland Garros and 20th career Grand Slam crown, will face Bacsinszky after the Swiss 23rd seed beat unseeded Belgian Alison Van Uytvanck 6-4, 7-5 in the day's other quarter-final tie.
The American had lost the first set in each of her last three matches in Paris but on this occasion she meant business right from the off, breaking immediately to lead 2-0 on the central Court Philippe Chatrier.
The diminutive Errani, who was the runner-up at Roland Garros in 2012 and lost to Williams in the semi-finals a year later, broke back straight away but once more she could not hold her own serve and fell 3-1 behind.
Williams then saved another break point against her in the following game with a brilliant cross-court forehand and, with Errani struggling against her opponent's power, she broke again before serving successfully to take the first set in just 27 minutes.
The second set was tighter, going with serve until Williams broke in the seventh game to lead 4-3, and she then went on to clinch victory on her fourth match point in a marathon ninth game as the match concluded in just one hour and five minutes.
Williams, who has now won all nine of her meetings with Errani, was far too powerful for the Italian, producing 39 winners to just nine for her opponent and conjuring 11 return winners against Errani's weak serve.
"In the last three matches I lost the first set and today I knew I needed to play very well because she is very good here at Roland Garros so it was very important for me," said Williams.
Bacsinszky in Hingis's footsteps
Bacsinszky is through to her first Grand Slam semi-final after repelling a battling performance by Van Uytvanck on Court Suzanne Lenglen in a match that lasted an hour and three quarters.
The Swiss got off to a great start, breaking twice early on to lead 3-0. Van Uytvanck then got one break back but could not find another and Bacsinszky took the opening set 6-4.
Van Uytvanck, the unseeded world number 93 who was the first Belgian to appear in the quarter-finals at Roland Garros since Justine Henin won the title in 2007, kept fighting and an exchange of breaks ensured that the second set was poised at 5-5.
But that was when Bacsinszky broke again, leaving her serving for the match. And the 25-year-old daughter of Hungarian parents converted her first match point with a terrific backhand down the line.
"It is absolutely extraordinary, just incredible. I'm very proud of my run," said Bacsinszky after becoming the first Swiss woman to reach the semis in Paris since Martina Hingis in 2001 and the first at any Grand Slam since Patty Schnyder at the Australian Open in 2004.
"It was a very difficult match. She was in the quarters because she had played very well so I had to battle like crazy, back to the wall."
Bacsinszky, who beat fourth seed Petra Kvitova in the fourth round, has lost both of her previous meetings with Williams, on clay in Rome in 2010 and in straight sets in the quarter-finals at Indian Wells earlier this year.
Williams is the heavy favourite to win the title, with the other semi-final pairing seventh seed Ana Ivanovic, the Serbian who was the French Open champion in 2008, with Czech 13th seed Lucie Safarova.