Six-time Rome champion Djokovic initially looked in improved form compared to his sluggish tournament opener against Tomas Etcheverry but again had a fight on his hands against the former world number three.
No longer wearing the elbow brace he had in Friday's second-round match, Djokovic looked to have shaken off the injury which forced him to miss last week's Madrid Masters.
The 22-time Grand Slam title winner lost his serve in the first game of the match but broke back immediately and in game four broke again to tip the first set in his favour.
Djokovic served out to take the lead and in game three of the second set broke former world number three Dimitrov, who handed the initiative to the Serb with a weak backhand into the net.
However, Dimitrov levelled the set at 4-4 with a break of his own, angering Djokovic who was convinced that the shot which won the game was long, and drew huge roars from the Foro Italico crowd when he claimed the set.
However, Djokovic has reached at least the quarter-finals in each of his 16 appearances in Rome and a powerful third-set display ensured his passage into the fourth round.
It was his 11th win in 12 meetings with Dimitrov.
"I was very happy to find the right rhythm in the first game of first set... That was super important to make a break early and hold things under control," said Djokovic who will face Britain's Cameron Norrie for a place in the quarter-finals
Swiatek breezes through
Losing finalist in Madrid last weekend, Swiatek looked again in imperious form in the tournament she won last year before going on to blitz the French Open.
The pole is trying for her third straight Rome title and made light work of Tsurenko after a slightly bumpy start.
Swiatek lost the first two games but then rattled off 12 in a row in a powerful demonstration of why she is world number one and favourite for Roland Garros.
"I'm just trying to be consistent and trying to play my game. I treat every match as a final, I try to have the right mindset," said Swiatek on court.
Swiatek now faces Donna Vekic, who saw off Ljudmila Samsonova 2-6, 7-6 (7/5), 6-2, after extending her winning streak in the Italian capital to 13 matches.
Her bid for a Rome hat trick will be helped by six of the world's top 10 women players being eliminated early.
Fifth-ranked Coco Gauff of the United States was defeated by Marie Bouzkova of the Czech Republic 4-6, 6-2, 6-2. It was Bouzkova's third career Top 5 win but first on clay.
Also heading for the exit was world number eight Maria Sakkari who was beaten 7-5, 6-3 by Marketa Vondrousova.
Gauff and Sakkari join world number two Aryna Sabalenka, third-ranked Jessica Pegula, world number four Caroline Garcia and seventh-ranked Ons Jabeur in being dumped out of the tournament.
Former world number one Victoria Azarenka withdrew with a right leg injury handing Madison Keys of the United States a free pass into the last 16.
Medvedev's Rome joy
Daniil Medvedev came through in straight sets against Emil Ruusuvuori to claim his first-ever win at the Foro Italico courts.
The former US Open champion won a second-round match which was postponed to Sunday due to the torrential rain which hit the Italian capital on Saturday.
In much balmier conditions the world number three was dominant on a clay court surface he is famous for not liking.
Few would back him to go deep in Rome but his performances on clay have improved, reaching the quarter-finals at the Monte Carlo Masters.
"Every time I come to a tournament I know I can play well, so I'm happy to finally get the win here in Rome," said Medvedev.
Medvedev will face Bernabe Zapata Miralles in the third round.
Fifth-ranked Stefanos Tsitsipas will take on home hope Lorenzo Sonego after finishing off Nuno Borges 6-3, 6-3 in a match interrupted by Saturday's rain with the Greek already well on his way to victory.
World number seven Holger Rune eased past Italian veteran Fabio Fognini 6-4, 6-2 to make the last 16 but eighth-ranked Jannik Sinner kept home hopes alive by seeing off Russia's Alexander Shevchenko 6-3, 6-7 (4/7), 6-2.
