0119 GMT March 30, 2020
The 30-year-old British world number one, whose victory in Rome last year was one of nine titles he won in 2016, lost 6-2 6-4 to the world number 29, BBC reported.
The loss continues Murray's poor form ahead of the French Open, which gets under way on May 28.
Murray's fellow Brit Aljaz Bedene was also knocked out in the second round by world number two Novak Djokovic.
The Serb, who has never failed to reach the last eight in Rome, dominated the tie-break to win a tight first set but eased through the second to win 7-6 (7-2) 6-2.
"A little bit of a slow start, but Bedene is the kind of player that gives you good rhythm," said Djokovic, who was beaten in the Madrid Open semis by Rafael Nadal last week.
"I had some good exchanges, some good games with rallies and it felt right, especially in the second set."
Murray, who turned 30 on Monday, continues to struggle for consistency on his return from an elbow injury.
He has won one event this season –on the hard court in Dubai in February– but has struggled on clay, with his best performance in the four events he has played so far on the surface being his semi-final appearance in Barcelona.
The Scot was under pressure from the very start, and failed to recover from losing his opening service game as home favorite Fognini swept into a 3-0 lead before closing out the set with a love service.
He was up against it again as more poor service games left him trailing 4-1.
There was a brief recovery by Murray as a break and a hold saw him trail 5-4 but Fognini reasserted his dominance to serve out victory and secure his first win over a world number one.
Murray's seventh defeat of the season–and his fifth in the last 10 matches–leaves him very short of confidence and form heading into the French Open.
Fognini hit some monster forehands, and some gorgeous drop shots, but at no stage was Murray able to impose his game on the Italian. Many of his groundstrokes were landing in mid-court: there was very little threat or conviction to trouble someone playing as well as Fognini.
Ivan Lendl flies to Europe this weekend to bolster Murray's coaching team, and they will all have their work cut out. Murray is currently playing nothing like a world number one, and nothing like a potential French Open champion.