France's Gasquet moved on to second round with a 6-2, 7-6 (10-8) victory over American Sam Querrey on the clay courts of the Foro Italico.
He'll next face Dimitrov, who triumphed over Cypriot Marcos Baghdatis 6-3, 6-4 at the Masters tennis event.
Other Day 1 victors included Uzbekistan's Denis Istomin, who recovered for a 5-7, 7-6 (7-5), 6-3 decision over American John Isner, and Ukrainian Alexandr Dolgopolov, who dispatched Italian wild card Matteo Viola 6-4, 6-3.
Jordana Brewster on Paul Walker: 'He was an enormous presence in my life'
Exploding whale video goes viral on Internet