The three times Wimbledon champion survived two set points, one with Young serving at 5-4 and the other on his own serve in the tiebreak at 5-6, before clinching victory with his fourth match point after the American double-faulted.
The Serbian world number four will play Russian Daniil Medvedev, who beat American fourth seed Steve Johnson 6-3 2-6 6-3, on Friday for a place in the final of the south coast grass tournament.
Top seed Djokovic - the highest ranked male player to compete at Eastbourne since 1999 - was the first through to the semis after an hour and 35 minutes on court.
"I enjoyed it, especially in the second set," he said. "The first set went my way and I played well. I felt good on the court and had some break point opportunities early in the second set.
"He served for the set, had a set point and then had set point in the tie-break ... Obviously, it could have gone easily his way. But it hasn't, and I'm just glad the way I kind of held my composure, my nerves.
"This is the kind of match situation that I was looking forward to having, and I'm glad it happened today and I managed to overcome that."
Second seed Gael Monfils beat British wild card Cameron Norrie 6-3 6-2 to reach the quarter-finals and then secured his place in the last four with a 7-6(4) 6-0 victory over Australian Bernard Tomic.
That set up an all-French semi against seventh seed Richard Gasquet, who beat South African Kevin Anderson and then third seeded American John Isner in quick succession.
(Reporting by Alan Baldwin; Editing by Alison Williams)
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