The Swiss world No20 rallied back from a set down to upset Radwanska 2-6, 6-4, 6-2 in two hours and two minutes, and make the last eight in the Premier Mandatory event in Florida for the first time.
Radwanska, a champion in Miami in 2012, showed her signature variety to race to a one set lead, hitting a mere seven unforced errors.
The Pole trailed by a break in the second and despite breaking Bacsinszky back when the Swiss was serving for the set at 5-3, she surrendered her serve once again and lost her advantage.
Bacsinszky did not look back from then on, benefitting from a slew of Radwanska errors to wrap up her first-ever top-three victory.
“Even when I wasn’t playing quite well in the first set, I told myself, it might be a long match,” said Bacsinszky, who last year made her top-10 debut in October.
“So I always tried to get a little more balls in and make her work.
“But I started to calm down a little bit – in the first set I was kind of nervous.”
Bacsinszky, who had taken out Ana Ivanovic in the previous round next faces world No5 Simona Halep, who got past an in-form Heather Watson 6-3, 6-4 in the last 16 Monday.
Halep dropped her first two service games against Watson to trail 1-3 but rallied back, winning the next five straight games to take the opener.
Great Britain’s Watson, currently ranked 69th in the world, was then broken again in the fifth game of the second set before breaking back to make it 3-3.
But yet another break put Halep 4-3 up before the pair held serve to give the Romanian the chance to serve for the match. The 2014 French Open finalist then wrapped up the win with a love service game to advance into the quarter-finals and take her record against Watson to a clean 3-0.
Halep’s best result in Miami was making the semi-finals last year.
The American top seed fell 6-7 (3), 6-1, 6-2 to Russia’s Kuznetsova in a 122-minute battle to lose prior to the quarter-finals in Miami for just the second time in her career.
Williams, who was targeting a remarkable fourth consecutive Miami crown and ninth overall, has not won a trophy so far this season and enters the clay court stretch title-less for the first time since 2012.
After splitting the first two sets, Williams went down a double break 0-3 in the decider.
She got one of the breaks back but Kuznetsova regained the advantage and closed out the match on a long ball from Williams, the 50th unforced error of the clash from the American.
“It’s obviously disappointing, but I’ve won here a lot, so it’s okay,” said Williams, who struck nine double faults.
The 34-year-old, who is based in Florida and trains there regularly, did not blame the heat and tough conditions for her surprise exit but admits her movement was sluggish.
Asked to elaborate on why she was unable to bring out her best, Williams said: “I don’t think it’s
appropriate to criticise my movement right now. I did the best that I could. I can’t win every match. The players come out and play me like they’ve never played before in their lives. You know, the best that I could do today. I have to be 300 per cent every day, so…”
Kuznetsova, who is now 3-8 lifetime against Williams apologised to the crowd for taking out their home favourite. “I’m sorry to the fans who are disappointed that Serena is not going to keep playing but for sure she will have more wins here,” said Kuznetsova.
Former World No 1 Rafael Nadal had to retire from his second-round match against Damir Dzumhur due to illness, most likely caused by the humid conditions at the Miami Open.
“It’s tough for me because I felt that I was playing well,” the Spaniard explained. “It’s a hard but that’s life.”
“I was playing well so I am in good form. If I can practice again in a couple of days I hope to be ready for the clay (season).”
“Even with such bad conditions I have been able to finish matches in my career because I did not want to retire like at the Australian Open a couple of times, but today my health was suffering.”