He took four hours but triumphed by 5-7, 6-2, 6-1, 7-6 (9-7) and will now face either Novak Djokovic or David Ferrer.
Their semi-final was beginning on Saturday night but may not finish, and it is still not clear if the final will take place on Sunday.
Today's play was delayed for an hour and a quarter by a storm and, although the rain cleared, the strong wind remained.
Murray went into the match as the favourite by ranking and experience at the latter stages of grand slams but also knowing Berdych had won four of their six previous matches.
The big-hitting Czech had also served notice of what a threat he would be by blasting past Roger Federer in the quarter-finals.
The wind was certainly the winner in the early stages as the stands slowly filled up and planes roared overhead after taking off from nearby La Guardia airport.
Murray was just pleased to be back on Arthur Ashe after his struggles on Louis Armstrong against Feliciano Lopez and Marin Cilic.
Murray piled the pressure on Berdych in a lengthy third game and got the break when the sixth seed drilled a forehand just wide.
Murray thought he had saved a break point in the next game but Berdych protested he had been distracted by his opponent's hat blowing off and, after a heated exchange between the two men and umpire Pascal Maria, the point was replayed and the Czech broke back to 2-2.
Berdych was struggling with his ball toss to such an extent that at times he was serving at little more than 80mph but he was certainly hitting the ball very cleanly off the ground.
The 26-year-old was two points from the set when he had Murray at 0-30 in the 10th game but the Olympic champion held on.
Holding serve into the wind was the main problem and Murray was furious with himself that he did not put more pressure on his opponent in the next game.
That left him serving to stay in the set once more and this time he could not manage it, the Scot struggling to keep his cool as Berdych broke through with a mighty swipe of his forehand.
So bad was the forecast that the women's final, scheduled for Saturday night, had already been postponed, and Murray and Berdych were making extremely slow progress.
The match had already been going for an hour and 20 minutes when they began the second set, but Murray was clearly fired up and immediately had three break points, taking the second when Berdych pushed a forehand long.
The Scot began the fourth game with a serve that died in the wind before Berdych could reach it, no doubt bringing a smile to the lips of club players everywhere, who would be much more familiar with such scenarios.
Murray seemed to be enjoying himself a bit more now, taking on the challenge instead of being frustrated by it, and he broke again to lead 4-1.
Berdych held to force the third seed to serve out the set, but no sooner had Murray thrown the ball up for his first serve than a gust of wind blew his chair and racquet bag across the court.
The Czech laughed at the impossibility of playing tennis in such conditions but on they went and Murray duly wrapped up the set 6-2.
Murray, who was being watched by Sean Connery, began the third set as he had the second, with a break of serve, as Berdych shot an exasperated look towards his supporters.
Murray's greater variety and tactical acumen suited the conditions much better than the Czech's raw power and another break for 3-0 gave him a firm hold on the match.
Mentally, Berdych seemed to have gone walkabout and, although he saved four set points at 1-5, a fifth arrived and this time the Czech drilled a forehand just long.
In the fourth set, Murray was cruising at 3-0 but allowed Berdych to break him and the set went to a tie-break which the Czech initially dominated, moving to 5-2.
But Murray dug deep and took his second match point to reach his second US open final.
