Marin Cilic had earlier dumped out a struggling Novak Djokovic to pave the way for Murray to supplant the Serb at the top.
But Murray was forced to save seven set points in an extraordinary opening tie-break before clinching a 7-6 (11/9), 7-5 win in just under two hours.
The Briton will meet Milos Raonic in semi-finals knowing that a victory will move him above Djokovic, who has been on top of the world rankings for 122 weeks.
“Before the match there were a few more nerves maybe than there was earlier in the week. But once I got out there, I didn’t feel any different to any other match,” said Murray.
“If it happens this week, great. But if not, I’m not putting any extra pressure on myself this week because I still think I have an opportunity to do it in the future.
“But obviously I’d love to do it tomorrow if I can.”
Neither Murray nor Berdych showed any signs of fragility as the first set went on serve, but the wheels fell off in spectacular fashion for the Czech after he raced 6-1 ahead in the tie-break.
Murray chalked up six straight points to earn a set point of his own, with Berdych then squandering two more before the second seed finally edged ahead. He broke Berdych to start the second set and looked on course to wrap things up, but a lapse at 5-4 saw him require a second chance.
Berdych, who needed to reach the last four to stay in the race for the Tour Finals in London, handed him that opportunity by again losing his serve, with Murray sealing it on his fourth match point.
– Cilic breaks duck –
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Cilic had lost all 14 previous meetings with Djokovic but snapped that miserable run with a 6-4, 7-6 (7/2) victory.
Djokovic, the record four-time champion, arrived in Paris needing to reach the final to ensure he remained ahead of a rapidly approaching Murray.
But the Serb’s bid to register a fourth straight title in the French capital ended abruptly as an improbable escape act fizzled out against an in-form Cilic.
“Credit to Marin and congratulations. He definitely played better today, and he deserved to win,” said 29-year-old Djokovic.
“I wasn’t on the level that I could have been on.
Obviously, there are things I could have done better. And just in important moments I wasn’t able to deliver.”
Ninth seed Cilic, who on Thursday became the seventh London qualifier, broke immediately to signal his intent against a player who had totally dominated him in his career.
In typical fashion Djokovic then hit back straight away to level, but Cilic’s confidence was undimmed and the Croat capitalised on a shaky service game from his opponent at 5-4 to grab the opening set.
Former US Open champion Cilic threatened again early in the second, but Djokovic landed a crucial breakthrough at 4-all to serve for the set.
The top seed uncharacteristically gifted the break back, though, after a pair of costly double faults, and Cilic had him on the ropes as two match points passed by at 6-5 on a faltering Djokovic serve.
There was to be no remarkable recovery either for Djokovic, as Cilic took charge in the tie-break and closed out a famous win at the fourth attempt.
“It’s good for me, I played great tennis. He had beat me 14 times but we had close matches and that helped me play better,” said Cilic.
PTI