The Scot, who twice served for the match, was left agitated at his Italian opponent as the contest neared its conclusion, accusing Fognini in a lengthy exchange at the net of making a noise to put him off during a rally.
"I've never had that in a match and I've played 800-odd matches on the tour," Murray said in a post-match news conference.
"Someone made a noise, I didn't know who made the noise, I looked in the direction of where the noise came from.
"He then told me, 'Stop looking at me, what are you looking at me for?'
"I was like, 'I was just about to hit a shot and someone made a noise.' He then told me to stop looking at him.
"He told me to stop complaining, to have a sense of humour. I wanted to know where the sound came from and it came from him, which you're not allowed to do.
"It's against the rules, it's hindrance, you shouldn't do it. But he said I should have a sense of humour about that, but in that moment neither of us were in a joking, laughing kind of mood.
"He [umpire Fergus Murphy] wasn't saying anything to him. I was obviously frustrated with that. He wanted to engage with me, I probably shouldn't have done, but I'm not having him talk to me like that on the court."
Murray's serve was broken early in the match, but the 32-year-old immediately responded to level the first set at 2-2. The Scot then pushed on for a 4-2 lead before his opponent erased the deficit with another break of serve, followed by a hold to love.
The set was ultimately decided in a closely-fought tiebreak, with Fognini emerging victorious.
The second set began with a mammoth game, which eventually ended with Fognini holding serve, and a deadlock ensued for several tense games before Murray broke for a 3-2 lead. This time, the former world No. 1 was able to surge clear and wrap up the set 6-2 to force a decider.
The final set followed serve until 4-4, when Murray found what appeared to be a crucial break to serve for the match. Fognini broke back, but immediately surrendered his own serve, allowing his opponent another chance to serve out the contest. Murray again faltered, however, setting up another tiebreak.
Fognini raced through the breaker to finally secure victory after more than three hours on court.
It was announced on Tuesday that Murray -- who reached the quarterfinals in Beijing last week before falling to top seed Dominic Thiem -- would make his Grand Slam return at next year's Australian Open.
The three-time major champion's last Slam appearance came at this year's Aussie Open, where Murray is a five-time runner-up. He fell to Roberto Bautista Agut in the first round in a five-set thriller, which the Scot thought at the time could be his final ever match.
Successful hip-resurfacing surgery enabled Murray to make his comeback to tennis this summer, however, with a period of doubles -- including a title at Queen's Club with partner Feliciano Lopez in Murray's first tournament back -- soon followed by his singles return.