Djokovic, who is ranked fourth in the world, took advantage of a double fault by Young late in the second set to dispatch his foe in one hour, 35 minutes. Djokovic is competing at Eastbourne for the first time as he prepares for Wimbledon, which begins on Monday.
"I enjoyed it, especially in the second set," Djokovic 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. ... I think Donald started playing with less unforced errors, playing with more variety and using his forehand very well from all over the court.
"He served for the set, had a set point and then had set point in the tie-break. It was very close second set. 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 situations that I was looking forward to having, and I'm glad it happened today and I managed to overcome that."
Djokovic improved to 26-7 on the season and will face Russian Daniil Medvedev, who posted a 6-3, 2-6, 6-3 victory over fourth-seeded American Steve Johnson.
The 21-year-old Medvedev lost only two points on his serve in the final set.
Medvedev had advanced to face Johnson after recording a 4-6, 6-1, 6-2 triumph over Robin Haase of the Netherlands.
Second-seeded Frenchman Gael Monfils recorded eight aces to complete a 6-3, 6-2 victory over local wild card Cameron Norrie of Great Britain. The contest had been resumed at 6-3, 2-1 following rain on Wednesday.
Monfils continued his impressive play with a 7-6 (4), 6-0 triumph in the quarterfinals over Australian Bernard Tomic, who earlier had defeated sixth-seed German Mischa Zverev 6-3, 6-2.
Monfils advances to face seventh-seeded Frenchman Richard Gasquet, who saved all three break points he faced to notch a 6-4, 6-4 victory over South African Kevin Anderson, 6-4, 6-4. Gasquet kept up his good fortune by breaking serve four times to breeze to a 6-3, 6-2 win over third-seeded American John Isner.
Isner gained entry into that match after claiming a 3-6, 7-6 (2), 7-6 (4) decision over Serb Dusan Lajovic in their postponed match.