I also had control, more or less, of a high percentage of the points.
The top Italian player came out wearing a black shirt with a red lightning bolt design down his chest and he was energized from the start, grazing the flower beds lining the court to return Thiem's heavily angled shots and frustrating the Austrian with well-placed lobs and touch volleys.
Next up for Ramos-Vinolas is a meeting with Novak Djokovic, who eased past Nikoloz Basilashvili 6-4 6-2. "He probably stopped to serve that big a little bit, and I was a little bit more relaxed".
"At the beginning, I really think that he was serving big".
Maria Sharapova saved two break points in the third set to get past Dominika Cibulkova 3-6, 6-4, 6-2.
Fognini battled hard to reach the quarters, coming from 2-4 down in the opening set to beat German Peter Gojowczyk 6-4 6-4. "That was the way that I want to play".
The 16-time Grand Slam victor next takes on Canadian teenager Denis Shapovalov who dug deep in a 7-6 (7/3), 6-7 (5/7), 6-3 win over Robin Haase of the Netherlands. The Madrid loss was clearly an aberration and Rafa is definitely looking determined to head to Paris with an eighth title on claycourts of Rome.
Goffin got through when Juan Martin del Potro had to retire at 6-2, 4-5 with a left leg injury he sought treatment for earlier in the match. Djokovic, a four-time victor at Rome, had five aces in the match and will next face Spaniard Albert Ramos-Vinolas, who rallied past eighth-seeded American John Isner 6-7 (5), 7-6 (2), 7-6 (5) in two hours, 53 minutes.More news: European Union to Move Friday to Block US Sanctions on Iran
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Third seed Grigor Dimitrov did not have a birthday to remember as he fell 7-6 (4), 5-7, 4-6 to Kei Nishikori.
Defending champion Alexander Zverev scraped past Britain's Kyle Edmund 7-5 7-6 (11) after squandering seven match points, and claimed an 11th consecutive victory that set up a quarter-final clash with ninth seed David Goffin.
Halep needs to reach at least the quarterfinals or risk losing the top spot to Caroline Wozniacki.
In the women's event, number eight seed and five-time Wimbledon champion Venus Williams suffered a shock loss to Estonia's Anett Kontaveit.
Wozniacki faces 15th-seeded Anastasija Sevastova in a late match.
The Romanian, a finalist in Rome past year, is on her way to holding her top ranking as she next plays France's Caroline Garcia or American Sloane Stephens.
Fourth seed Marin Cilic displayed plenty of patience during his 6-3 6-4 win over Frenchman Benoit Paire.