Shapovalov, who will face Nadal on Thursday, finished with seven aces and converted three of 11 break point opportunities in a match that lasted two hours 26 minutes. Dimitrov, the world No. 4 who is celebrating his 27th birthday on Wednesday, also lost his first-round match at last week's Madrid Open.
"I played very well today", said Thiem.
Former world number one Novak Djokovic is also in the quarter-finals for the first time this year. But with my serve, I didn't lose many points.
The Spaniard will next take on Fognini, a player he knows well having played 14 times and holds a 10-3 record against including five wins in their most recent matches. "I hope in Paris I will have a little bit longer time to come into the tournament".
"At the beginning, I really think that he was serving big". 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.
Seeking an 8th Rome title, ahead will be Italian bad boy Fabio Fognini who bested Peter Gojowczyk 6-4, 6-4.
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. "I've historically always played well in Rome, and I feel like each match is getting better".
Although six seeded players suffered exits on Wednesday (May 16), Juan Martin del Potro and Marin Cilic were able to advance.
Shapovalov is 7-4 on clay this year, including a run to the semifinals at Madrid.
Defending champion Alexander Zverev of Germany kept his title defence on track but needed eight match points to see off Briton Kyle Edmund 7-5, 7-6 (13/11).
Halep plays US Open champion Sloane Stephens or seventh-seeded Caroline Garcia.
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.
Second round: Maria Sharapova bt Dominika Cibulkova 3-6, 6-4, 6-2; Venus Williams bt Elena Vesnina 6-2, 4-6, 7-5; Anett Kontaveit bt Svetlana Kuznetsova 7-5, 7-5; Caroline Wozniacki bt Alison Van Uytvanck 6-1, 6-4; Daria Gavrilova bt Garbine Muguruza 5-7, 6-2, 7-6 (6).
The Romanian will next play 13th-seeded American Madison Keys, who battled past Croatian Donna Vekic 7-6 (7/2), 7-6 (7/0) for a place in the quarter-finals.