The second-ranked Djokovic dropped his serve once in each set but managed to recover to beat Thomaz Bellucci 6-4, 6-4, while Spain's Verdasco eliminated compatriot illermo Garcia-Lopez 6-4, 7-6 (2) to take his record on clay this year to 13-2.
Djokovic will now look to avenge his defeat to Verdasco in the semifinals of the Monte Carlo Masters two weeks ago.
"I've played a lot of matches against him on different surfaces and I've won most of them," Djokovic said of his 5-3 career record against Verdasco, adding that the Monte Carlo match "wasn't a real picture of my game. I didn't feel good on the court and made a lot of unforced errors."
Seventh-seeded Jo-Wilfried Tsonga also advanced by beating Colombian qualifier Santiago Giraldo 6-3, 6-4, while Ernests Gulbis followed up his upset win over Roger Federer with a three-set victory over Italian wild card Filippo Volandri, 6-2, 3-6, 7-6 (4) - fighting off a partisan crowd in the process.
The 21-year-old Gulbis will next meet Spain's Feliciano Lopez, who advanced when 11th-seeded Ivan Ljubicic appeared to aggravate a muscle problem in his side during the pre-match warmup.
Against the 28th-ranked Bellucci, Djokovic got in just 57 percent of his first attempts - although he committed half as many unforced errors as the Brazilian player.
"I was down a break every set and it wasn't easy to get back on serve," Djokovic said. "He served well - very accurately - and got a lot of free points." Djokovic is aiming to reach his third consecutive final in Rome, having won in 2008 and lost to Rafael Nadal last year.
Later, Nadal was up against No. 39 Victor Hanescu in the night match at the Foro Italico.
In Stuttgart, top seed and world No. 2 Caroline Wozniacki crashed out of the WTA tournament here on Thursday after suffering from a recurring ankle injury, just three weeks before the French Open.
The Danish teenager lost 6-4, 6-4 to Lucie Safarova of the Czech Republic in just 66 minutes of her second-round match as the ankle injury she suffered in Charleston earlier this month flared up again.
"I didn't feel like I could move 100 percent and if you can't move, you can't run," said the 19-year-old, who withdrew from her semi-final in Charleston with a sprained ankle.
"She was making me run a bit, but now I just want to get ready for my next tournament (in Madrid)." It was Wozniacki's first match since her American withdrawal and she never looked comfortable as Safarova failed to allow her even a single break point while the Czech player won 83 percent of her first serve points.
"I was a bit nervous before, but I saw Caroline wasn't moving so well (because of her ankle injury)," said Safarova.
"I really wanted to win this match and I just focused on myself.
I worked hard to keep my serving and keep her under pressure and it paid off." Safarova will now play qualifier Anna Lapushchenkova in Friday's quarterfinals after the Russian defeated sixth seed Victoria Azarenka 6-3, 6-3.
Fourth seed and former world No. 1 Jelena Jankovic is also in the quarterfinals after beating Bulgaria's Tsvetana Pironkova 6-2, 6-2 in just 82 minutes.
"Overall, I played well," said Jankovic. "I was aggressive, I went after my shots and I served well, there is always something I could have done better, but I am happy overall."
The 25-year-old admitted to feeling tired and says she is turning to German drink apfelschorle - apple juice and sparkling water - to help her as she prepares to face either Yanina Wickmayer or Justine Henin in the last eight.
"Either way, it will be a tough match, I hope for the best," said Jankovic.
Australia's Samantha Stosur survived a first set scare before she rallied to beat Romania's Alexandra Dulgheru 3-6, 6-0, 6-2 in her second-round match.
The seventh-seed looked in trouble in the first, but roared back in the second and third to blow away her opponent and set up a quarterfinal clash with China's Li Na on Friday.
