The usual guidelines about the invincibles and the journeymen are set aside in the Davis Cup.
The 17-time Grand Slam winner Williams complained about being mentally fatigued after the 84 minute contest.
India No. 2 Bhambri pulled out of the tie last week after injuring his left heel while training in Delhi.
Switzerland last reached a Davis Cup semi 11 years ago, when Federer lost a key five-setter against Australia’s Lleyton Hewitt.
Despite Sunday’s loss, Nadal continues to hold a slight edge, 22-18, in their head-to-head meetings.
Djoko blasted five aces, won 30 of 35 points on his first serve and had no double faults in the 83-minute contest.
Number one Williams and number two Li have both won two titles each this season and sit atop the world rankings.
Nishikori has been one of Federer’s bogeymen, having now gotten the better of him twice in three meetings.
21st ranked Nishikori defeated 17-time grand slam champion Federer 3-6 7-5 6-4 on Wednesday.
World No. 1 Nadal brushed past Italy’s Fabio Fognini to join Djokovic, Federer and Murray in last eight.
Andy Murray and Novak Djokovic will face each other in the quarter finals of Sony Open.
Serena's triumph against fifth-seed Kerber came at a time when the complex was locked due to a bomb scare.
Sharapova endured her second successive ragged three-setter and found herself trailing 4-0 after only 15 minutes.
Murray polished his record to 9-0 over Lopez in only 73 minutes. The Scotsman improved to 21-6 in the US city.
Rafael Nadal has won 62 titles but a victory on the Miami hardcourts is not among them.
Nadal pounded his way to a one-sided 6-1, 6-3 win over Hewitt.
I felt like I did the best that I could and I’m happy with the way I’m coming back, said 21-year-old Tomic.
The top-ranked Williams seeks a record seventh Key Biscayne title, and her second in a row.
The tournament began on a down note for the home contingent with American Bethanie Mattek-Sands falling 6-3 3-6 6-3 to Ukraine’s Elina Svitolina.
Murray will be defending his Miami title next week before leading Britain in the quarters of Davis Cup next month.
Federer moved up three spots to No. 5 in Monday's rankings. That's the closest he's been to top four since last June.
Federer still leads the series 17-16, having beaten Djokovic in three sets in the semifinals at Dubai two weeks ago.
The Swiss master has already won the Indian Wells title four times and made it through to a fifth final.
The Swiss master succeeded where others had failed by blunting the serve of the towering 6ft 8in (2.13m) Anderson to win 7-5 6-1.
Federer succeeded where others had failed by blunting the serve of the towering 6ft 8in (2.13m) Anderson.