However, Shapovalov recovered to hold and saved five more break points in his next service game to keep the set on serve. Nadal lost their only other match at the Rogers Cup in Montreal last summer.
Former world number one Novak Djokovic is also in the quarter-finals for the first time this year.
"It's true that I had 15-40 and a couple of break points in the first game, 15-40 and a couple of break points in the second return game and I didn't make them". However, Madrid finalist, Dominic Thiem was sent packing by Italy's Fabio Fognini in three sets 6/4, 1/6, 6/3.
"We're finally connecting and I think that's the most lovely thing", Fognini said of the crowd, which whistled him off the court following an erratic performance a few years ago.
"It was important, after a loss in Madrid, to come back strong, and that's what I did today", said Nadal following his demolition of Dzumhur.
Putin Met With Assad in Sochi - Kremlin Spokesman
But rebel factions still control large parts of northwestern and southwestern Syria, while a Kurdish-led alliance backed by the United States holds most of the northeast.
Nadal will next face Canada's Denis Shapovalov, who faced a grueling test in battling past Dutchman Robin Haase 7-6 (3), 6-7 (5), 6-3 in almost two and a half hours.
Nadal, who hasn't won the Italian Open since 2013, next faces Fabio Fognini, who beat Peter Gojowczyk 6-4, 6-4. "[Fognini] is a tough player and I'm looking forward to the challenge".
Shapovalov is 7-4 on clay this year, including a run to the semifinals at Madrid.
Defending champion Alexander Zverev won his 11th straight match but had to work really hard to beat Kyle Edmund needing eight match points in the second set tiebreaker to win it 7-5, 7-6 (11).
Halep needed to reach at least the last eight and go further than world number two Caroline Wozniacki to retain the top-ranking going into the French Open.
IPL 2018: RCB thrash KXIP by 10 wickets, remain in playoff race
KXIP is concern about Captain R Ashwin's inconsistent form, has picked only six wickets in 11 matches with 8.13 runs per over. Back-to-back wins for the first time in two seasons has given RCB's quest for a maiden IPL title a fresh lease of life.
The 22-year-old Kontaveit won through 6-2, 7-6 (7/3) against the 37-year-old American who was seeded eighth in a tournament she won almost two decades ago to 1999.
The Romanian, a finalist in Rome a year ago, is on her way to holding on to top spot as she next plays France's Caroline Garcia who dumped out US Open champion Sloane Stephens 6-1, 7-6 (9/7).
The 21-year-old will play Belgian David Goffin after the ninth seed advanced when Argentine fifth seed Juan Martin del Potro retired while trailing 6-2, 4-5 with a groin injury.
Cilic faces 10th seed Pablo Carreno Busta, who beat Aljaz Bedene 6-4 6-7 (7-3) 6-2.
Svitolina, the fourth seed, received a first round bye and dropped just three games in a dominant 6-1, 6-2 victory over 35th-ranked Petra Martic.
Microsoft plans lower-cost Surface tablets to battle Apple
The new cheaper Surfaces could likewise appeal to students and teachers, and to schools that buy less-expensive devices in bulk. Microsoft may have decided it was time to change that; the rumored tablets would upgrade previous offerings in a big way.
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