Both players recorded double figures of break points, but it was the teenager often likened to Rafael Nadal who made the most of his opportunities to get a lead over Murray.
Alcaraz converted five of 15 break point opportunities, while the former world No 1 won just two of his 10 break points.
He managed to improve his first serve percentage by the end of the match, making 62% of his serves compared to Murray’s 51%, and winning 59% of the points during the two-hour battle.
The pair also enjoyed a lengthy encounter in Indian Wells earlier this month, as Murray called it a “brutal match”.
“Three hours for three sets is a bit much,” he said, after sealing a 5-7 6-3 6-2 victory over Alcaraz in California.
Unable to get the better of the 18-year-old once again, Murray will now look to the upcoming Paris Masters – where he says he is open to playing the qualifying event – and the ATP 250 in Stockholm.
After ruling himself out of the Davis Cup, he also admitted he could be convinced to play and the British squad remains one player short.