The reigning champion had not dropped a set en route to this last-four meeting in Doha, but saw his title defence hanging by a thread as the dangerous Verdasco recovered from 4-2 down in the opening set.
The Spaniard reeled off four games in succession to get in front and was seemingly on the brink of an upset when he took a 6-2 lead in the second-set tie-break.
But Djokovic showed his customary steel to save all four match points and then another at 6-7 down, three successive points levelling the match.
Those missed opportunities perhaps knocked the stuffing out of Verdasco, who was unable to make any inroads on the Djokovic serve in the decider.
The Serbian, meanwhile, broke in games five and nine to seal a remarkable comeback and set up a meeting with world number one Murray, who took an easier path to progress against Tomas Berdych.
Murray prevailed 6-3 6-4 against the 2015 finalist, running down a Berdych drop-shot for the crucial break in game four of the first set, sealed with back-to-back aces.
The start of the second set saw both men struggle to hold serve, although it was Murray who got his nose in front at 3-2.
The Scot wasted two match points when serving for victory, but he capitalised on a third to set up a mouth-watering meeting with his old foe.
Meanwhile, in the Chennai Open the four semi-finalists were decided. Roberto Bautista Agut will play Benoit Paire, and Daniil Medvedev will take on Dudi Sela.