American Paul, 24, recovered from a second-set wobble to see off the former world number one 6-2 3-6 6-3 in two hours and 15 minutes.
Murray had produced one of his best displays since undergoing hip surgery in early 2019 to defeat top seed Jannik Sinner in the previous round, but was unable to maintain his form.
The 34-year-old Scot struggled to cope with the weight and depth of Paul's groundstrokes as he was outplayed in the opening set, twice losing serve, in 45 minutes.
Paul, who knocked out fifth-seeded compatriot Taylor Fritz in the second round, was playing at the top of his game and Murray was forced to work hard on his serve to edge 2-1 ahead in the second set.
At 2-2, the younger man's clean hitting was unrelenting, but two brilliant trademark lobs helped edge Murray back in front at 3-2.
Murray was further rewarded for his patience by breaking to go 4-2 up after Paul's level dipped in the sixth game.
That became 5-2 when the American missed a crucial half-volley at 30-30 in the next game and Murray went on to level the match, serving out to win the set 6-3 after an hour and 37 minutes.
In the fourth game of the decider, with Murray's serve under pressure, Paul missed a routine forehand volley for break point and the Scot went on to hold at 2-2.
Paul held serve more comfortably as a compelling duel unfolded, with Murray forced to call on all his resilience as the deciding set was locked at 3-3.
But at 4-3 down, Murray double-faulted – his first of the match – on the first break point of the set and at 5-3 up, Paul served out for the match to set up a last-four clash with eighth-seeded compatriot Frances Tiafoe. Canadian Felix Auger Aliassime qualified for the other semi-final, scheduled to face the winner of Denis Shapovalov’s quarter final with Denis Rinderknech.
While Murray’s season is now over, defeating Jannik Sinner in the last 16 earned his second win over a top ten ranked player in as many months, having defeated Hubert Hurkacz in Vienna last month.