The Spaniard, a seven-time champion in Rome, swept Djokovic aside in the semi-finals of the Madrid Masters last week before going on to secure the title.
Nadal extended his current winning streak on clay to 17 matches with a 6-3 6-4 win over American Jack Sock in their quarter-final.
Djokovic, beaten in the final by Britain s Andy Murray a year ago, will now face Juan Martin del Potro in the quarters after the unseeded Argentinian s 7-6 (7/4), 6-3 win over Japanese seventh seed Kei Nishikori.
Novak Djokovic has endured a disappointing year and recently parted company with his long-time coaching staff, but the Serbian appeared to have rediscovered some drive and broke three times to advance under blue skies in Rome.
"I am here to try my best".
Djokovic was moving around the court more freely than he has been for much of this season, breaking to lead 4-3 when Bautista Agut netted a backhand and wrapping up the opening set after saving a first break point in the next game.
Third seed Stan Wawrinka became the latest high-profile casualty at the Italian Open on Thursday when the Swiss was knocked out in the third round by American John Isner. "I am very proud and humbled".
"I played a more aggressive game and I m serving better".
The Austrian fought off a double match point at 4-6 in the tiebreaker, then thwarted a third match point at 6-7 before surviving.
But French Open champion Djokovic said: "I m getting closer, I feel like today s match was one of the better ones I ve played on clay this season, and in the past couple of months".
Isner's quarterfinal opponent will be another big server, sixth-seed Marin Cilic, who eliminated ninth-seeded David Goffin of Belgium 6-3, 6-4.
If one looks at the head-to-head record, Djokovic has been mighty impressive, winning 13 matches and only losing four.
Zverev, the 16th seed, will meet Canada's world No 6 Milos Raonic after the Wimbledon finalist comfortably defeated Tomas Berdych 6-3 6-2.
In the women's tournament, Venus Williams overcame determined Briton Johanna Konta to win their third-round clash 6-1, 3-6, 6-1.
Qualifier Daria Gavrilova beat seventh-seeded Svetlana Kuznetsova 2-6, 7-5, 6-4 and Kiki Bertens won 7-6 (3), 6-1 over Russia's Ekaterina Makarova.