Djokovic started strongly, holding his opening two service games to love but from 2-2 a pumped-up Murray then won seven games in a row to claim the first set and go 3-0 up in the second on a balmy evening in Dubai.
Murray served for the match at 5-3 and nerves got the better of the world number four, double-faulting as Djokovic, 24, earned two break points.
A netted backhand from the Briton then enabled Djokovic to pull it back to 5-4 and the set appeared to be heading for a tiebreak. However, Murray roused himself to break again and seal victory after the Serb clubbed a forehand into the net.
Murray, 24, will now play the winner of Friday's other semifinal between 16-times grand slam winner Roger Federer and Argentine Juan Martin Del Potro.
Only four points were dropped in the opening five service games, all by Murray, with the Serb seemingly assured as he sought to extend a 7-4 winning record against the Briton.
After appearing impregnable, Djokovic then made three errors in successive points while on serve, two from backhands that went wide and one forehand slugged into the net to give Murray a first break point.
Both players were cautious, scared of making an error on the match's first big point, with Murray in particular seeking to take the pace out of the rally with some sliced backhands.
Djokovic eventually lost patience and a wild forehand gave Murray the first break and a 4-2 lead.
Djokovic, seeking a fourth straight Dubai title, rallied and went 15-40 up in the next game but Murray saved both break points, the first at the net with a backhand drop shot and the second with a fierce drive to Djokovic's feet.
He eventually held for 5-2 and it was to get better for the Briton, Djokovic making more errors to slump to 0-40 down for three set points to Murray.
The three-times grand slam finalist took the second, a short Djokovic shot inviting Murray forward to send a backhand winner into the left corner for the first set.
Murray broke for a third time to go 2-0 ahead in the second set as Djokovic's errors racked up.
Facing two break points, Djokovic seemed in control of the first by working Murray from side to side as both players hugged the baseline, but the Serb snubbed the chance of a smash winner from deep court and eventually played a forehand long.
The set then went with serve to leave Murray with ball in hand for the match, but five-times grand slam winner Djokovic broke for the first time after forcing two successive errors from his opponent and squared for 5-5.
Yet Murray, who a day earlier said he had only ever lost once or twice after serving for the match, came again for his first victory over the Serb since 2009 - excluding a Djokovic retirement in Cincinnati last year.
