Murray responded to lead 0-30 in Federer's next service game, but two marvellous drop shots, an ace and stunning cross-court forehand put the world number one within a game of levelling.
But the Briton took the next four points, gaining a crucial one on the Frenchman's serve when he was quick to react to a drop shot and steer a forehand winner cross court.
Federer was struggling but he had his chances at the start of the second set, twice failing to capitalise from 15-40, with one routine mid-court forehand particularly surprising as he blazed it way over the baseline.
BBC: Andy Murray beats Roger Federer to win Shanghai Masters
In the end, it was Davydenko who brought the world number three back into the match with a poor game at 4-4, making a backhand error before going for too much on a cross-court forehand and dragging it into the tramlines on break point.
The set now back under his control, Murray closed it out with another break as he fired one brilliant winner down the line from deep behind the baseline, before racing in to collect a drop shot and whip a cross-court forehand past Federer on set point.
BBC: Andy Murray beats Roger Federer to win Shanghai Masters
An emphatic forehand winner to the deuce court gave Murray a 5-2 lead and he completed his victory with a running forehand put-away.
In Rome this year, Nadal attempted 15 first serves to Djokovic's forehand in the deuce court, compared to 10 down the center "T" to his backhand, according to Craig O'Shannessy, a strategy analyst who compiles data on pro tennis.
WSJ: Rafael Nadal, the Underrated Tactician, Stalks Novak Djokovic
Delbonis took the first two games easily, knocking Nadal around the court with a heavy forehand.
Lin is an aggressive all-court player, with a powerful topspin forehand honed by his childhood coach: his father Li, a professor of engineering at the University at Buffalo.
Golding saved three break points at 2-2 before seizing his single opportunity on the Olivo serve at 5-4, whipping a forehand winner past his opponent to the delight of the large Court 12 crowd.
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But Jankovic had just too much power as a huge cross-court backhand set up a fourth chance, and an equally heavy forehand winner put the former world number one into a seemingly decisive 5-3 lead.
Despite lacking consistency at present, 16-year-old Broady - a recent practice partner for Rafael Nadal - possesses an impressive physique, has a calm on-court demeanour, is light on his feet and benefits from a potent forehand.
He wasted a match point in game seven by hitting a forehand into the net, but eventually ended his opponent's challenge after one hour and 21 minutes on court with a neat drop shot.
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