Rose looked set for a remarkable third win in succession to seal a second Race to Dubai title when he led by a shot with seven holes to play, only to bogey three of the next five.
That left the Olympic champion needing to eagle the last hole to overhaul Fleetwood, but his long-range attempt slid past the hole as Fleetwood looked on, alongside his fiancée, Clare Craig, and their 7-week-old son.
"It's kind of amazing," Fleetwood said. "It's not kind of sunk in yet. It was great being out there today trying to win it. It really was out of my control over the last few holes.
"It was difficult being sat on the couch in the scoring area, just watching TV, not being able to do anything. You just have to watch how it unfolds and I felt for Justin a lot. I think how gracious he was shows a lot about his character.
"It's the biggest day of my career for sure. The achievement of winning a year-long accomplishment is massive and it holds a lot of respect amongst your peers. It shows the level of consistency and the amount I've improved as a player and as a person."
Rose had to settle for a tie for fourth at 17 under par despite starting the final round with a 1-shot advantage. It all started to go wrong for Rose on the back nine.
"I felt like I was in complete control until the 12th hole then I kind of hit the buffers," Rose reflected. "I don't know where it went wrong on the back nine.
"I'm happy for Tommy. He's battled hard all year. It's good for him to finish it off."
Rose finished 2 shots behind European Tour rookie of the year Rahm, 23, who secured his second victory of the season thanks to a closing 67.
"When you accomplish something in life, it's always a great feeling," said Rahm on his victory in Dubai. "When you do something that really wasn't a goal and you accomplish something that was years down the road, it's just such an incredible feeling.
"I wanted it really bad and I couldn't calm down going out to the course. Putts weren't dropping, but when I birdied 10, I'm like, 'OK, this is a good moment.'
"That was followed by about three or four horrible iron shots, but I managed to make pars, and when I made that birdie putt on 14 I was tied for the lead. I took advantage of a lucky bounce on 16, and thankfully that was enough."