American Paul, 24, recovered from a second set wobbling to overlook the previous world set 6-2 3-6 6-3 in two hours and 15 minutes.
Murray had produced one of his best performances since undergoing hip surgery in early 2019 to defeat top seed Jannik Sinner in the previous round, but could not retain his form.
The 34-year-old shooter struggled to cope with the weight and depth of Paul’s ground stroke when he was played out in the opening set, twice losing the serve, in 45 minutes.
Paul, who knocked out fifth-seeded compatriot Taylor Fritz in the second round, played at the top of his game and Murray was forced to work hard on his serve to lead 2-1 in the second set.
At 2-2, the younger man’s pure hit was relentless, but two brilliant brand lobbies helped Murray back to 3-2.
Murray was further rewarded for his patience by breaking to 4-2 after Paul’s level dropped in the sixth game.
It became 5-2 when the American missed a decisive half volley at 30-30 in the next match and Murray continued to level the game and won the set 6-3 after an hour and 37 minutes.
In the fourth game of the decision, with Murray’s serve under pressure, Paul missed a routine forehand volley for break points and the shots continued 2-2.
Paul held the serve more comfortably when a convincing duel took place, with Murray forced to call on all his resilience as the decisive set was locked at 3-3.
But at 4-3 under, Murray doubled – his first of the match – at the first break point of the set and at 5-3 up, Paul served out for the match to create a final four-stroke with eighth-seeded compatriot Frances Tiafoe. Canadian Felix Auger Aliassime qualified for the second semifinal, scheduled to face the winner of Denis Shapovalov’s quarterfinals with Denis Rinderknech.
While Murray’s season is now over, Jannik Sinner defeated in the round of 16 his second win over a top ten ranked player in as many months, after defeating Hubert Hurkacz in Vienna last month.