Spain's Robredo, seeded 16th in the US Open tune-up, on Thursday sent top-seeded Djokovic packing 7-6 (8/6), 7-5, the defeat coming a week after the world number one Serb was beaten in the third round at Toronto.
The Wimbledon champion now goes into the US Open, the final Grand Slam of the year which starts on August 25, well short of matches and confidence.
Second-seeded Federer, a five-time Cincinnati champion, held off Monfils 6-4, 4-6, 6-3.
He booked a tough quarter-final against eighth-seeded Scot Andy Murray, who battled more than two hours against the huge serve of John Isner, finally dispatching the American 6-7 (3/7), 6-4, 7-6 (7/2).
Federer had a rare show of temper late in the match, arguing with the chair umpire over a decision not to call on electronic line-calling on a particular point.
But Federer quickly recovered his poise, clinching the match with a service winner after breaking Monfils in the penultimate game.
"I'm just happy the way I'm playing right now," Federer said. "It was much better than the last couple of matches.
"I was hitting good forehands, not making so many mistakes, returning better," he added. "So there was a lot of positives out in the match. Sometimes frustration, but sometimes that's what Gael does to you."
- Djokovic 'not clicking' -
Djokovic, who was married last month and is soon to become a father, said it wasn't a question of the new elements in his life weighing on his tennis.
"I just lost the match," he said. "It was bad. Many, many things are not clicking these two weeks on hard courts. It's unfortunate, but it's more than obvious I'm not playing even close to what I'm supposed to play.
"I have to keep on working and trying to get better for the US Open. I just don't feel comfortable. That's it."
Robredo was thrilled to win only his second match in eight encounters with a world number one player. He beat then-top ranked Lleyton Hewitt at Roland Garros 11 years ago.
"It's amazing to be here playing and to beat the number one, I'm more than happy," said the 32-year-old Robredo, who next faces sixth-seeded compatriot David Ferrer -- a 7-5, 6-0 winner over Mikhail Youzhny.
Murray, in search of his first title since Wimbledon in 2013, has struggled to regain top form since his return from back surgery in January.
"It's a very important match for me to win," he said. "I had lost a few close matches over the last few months. It was important for me to come through."
Australian Open champion Stan Wawrinka, the third seed, rallied to beat Croatian Marin Cilic 3-6, 6-0, 6-1.
He'll play France's Julien Benneteau, who beat Jerzy Janowicz 7-5, 6-1.
Fifth seed Milos Raonic made amends with his 30th ace on match point after earlier double-faulting three times while serving for victory.
The Canadian defeated American Steve Johnson 6-7 (7/9), 6-3, 7-6 (7/4) and next faces Italian 15th seed Fabio Fognini. Fognini beat Lu Yen-Hsun of Taiwan 3-6, 6-3, 6-3.
- Sharapova, Halep book rematch -
In women's action in the combined ATP and WTA event, defending champion Serena Williams continued her march through the draw, beating Italian 13th seed Flavia Pennetta 6-2, 6-2.
Williams next plays eighth seed Jelena Jankovic, who beat Sloane Stephens 7-6 (7/4), 6-4.
Also through was world number two and second seed Simona Halep, a 6-4, 7-5 winner against Czech Lucie Safarova.
"Today was much better than my first match here," said Halep, winner in Bucharest last month and runner-up to Maria Sharapova at Roland Garros. "I was happy to win in two sets, since she was up 5-4.
Halep earned a mouth-watering rematch with fifth-seeded Sharapova, who beat fellow Russian Anastasia Pavlyuchenkova 6-4, 7-6 (7/2).
Sharapova had to rally from a break down three times in the second set but managed to finish it off in two.
Ukrainian teenager Elina Svitolina, who took out double Wimbledon winner Petra Kvitova in the second round, claimed her second seeded victim with a defeat of Spain's number 15 Carla Suarez Navarro 4-6, 6-4, 6-4.
She'll next face former world number one Ana Ivanovic, a 6-2, 2-6, 6-3 winner over Svetlana Kuznetsova.
Fourth-seeded Agnieszka Radwanska of Poland cruised past Germany's Sabine Lisicki 6-1, 6-1 and will take on Denmark's Caroline Wozniacki, a 7-5, 6-2 winner over sixth-seeded German Angelique Kerber.