It was Tipsarevic's second win in five matches against Djokovic, who last lost at this stage in November at the Paris Masters.
Nadal was beaten by Fernando Verdasco on Thursday. He later said he wouldn't return unless the tournament reverts to red clay.
The top-ranked Djokovic said the same.
"I want to forget this week as soon as possible and move on to the real clay courts," Djokovic said. "Here you can't predict the ball bounce or movement. They can do whatever they want, but I won't be here next year if this clay stays."
While Djokovic had little trouble holding serve early on, Tipsarevic saved four break points to force the first-set tiebreaker. The seventh-seeded Tipsarevic broke the defending champion early in the second set, closing it out after Djokovic had saved three match points.
Tipsarevic will now play Roger Federer and the Swiss star knocked out David Ferrer 6-4, 6-4.
The blue clay didn't stop the 16-time Grand Slam champion from deploying his usual array of unreachable shots to remain unbeaten in 13 matches with the sixth-ranked Spaniard, who didn't help his cause by serving seven double faults.
Federer struck back-to-back aces to clinch the win, improving his record to 24-3 this year.
Also, Juan Martin del Potro served 10 aces in beating Alexandr Dolgopolov of Ukraine 6-3, 6-4. The Argentine will play Tomas Berdych in the semifinals.
The six-seeded Berdych won 6-1, 6-2 over Verdasco, who made 24 unforced errors and showed little of the flair on display against Nadal.
On the women's side, Serena Williams brushed aside second-seeded Maria Sharapova 6-1, 6-3 to advance to the semifinals along with top-ranked Victoria Azarenka.
Williams opened strongly, serving six aces and earning two early breaks to take the first set.
Sharapova finally showed some fight and broke to love to level at 2-2 in the second set, but she quickly squandered her break with a costly sixth double fault to fall behind and ultimately lose to Williams for the seventh straight match.
"Playing Maria you have to be really ready because she does everything really well," said Williams. "I felt relaxed out there, and when I play relaxed I play better."
Williams, who improved to 11-0 on clay this year, will face Lucie Hradecka after the Czech upset U.S. Open champion Sam Stosur 7-6 (8), 7-6 (6).
Azarenka overcame a shaky start to beat French Open winner Li Na 3-6, 6-3, 6-3.
Azarenka struggled initially with her serve but broke her Chinese opponent's first two service games of the second set en route to levelling the match.
Azarenka, who has won four titles this season and was last year's runner-up in Madrid, used her powerful groundstrokes to jump out to a 4-0 lead in the decider before holding off Li's late charge.
"She was pretty unbelievable in the first set," said Azarenka. "I had to turn something around. I don't know if it was confidence or if it was self belief or just pushing myself to be better and really wanting to win the match."
Azarenka said she agreed with Nadal and Djokovic in their criticism of court's surface.
"You feel unstable sometimes. Actually, a lot of times," she said. "But right now there is no point on talking about it. After the tournament all the players can get together and discuss it."
The Belarusian will next face fourth-seeded Agnieszka Radwanska, who beat Varvara Lepchenko 6-4, 6-4.