Paul was down 6-2, 3-0 on the Hamburg clay but surged back to win 2-6, 6-3, 6-3 and set up a meeting with Peruvian qualifier Ignacio Buse in Saturday's final.
The American's comeback was kick-started when he broke de Minaur's serve for the first time on the 11th attempt, before winning nine consecutive games.
Paul converted his third match point to win in two hours and 17 minutes and qualify for the 10th final in his career -- and his third this season.
Earlier on Friday, Buse completed a 6-1, 6-4 win over Aleksandar Kovacevic to reach his first ATP tour final.
The world number 57, who knocked out defending champion Fabio Cobolli on Tuesday, roared out of the blocks and breezed through the first set in just 26 minutes.
The 22-year-old was again untroubled on his serve in the second set and wrapped up victory in one hour and four minutes.