Djokovic admitted he had to stay mentally tough as the three-time champion shrugged off a controversial call against him to beat Edmund.
The pressure is double on Kyle Edmund, the world number 17, as two-time Wimbledon champion Andy Murray withdrew on the eve of the tournament due to his long-standing hip injury.
Murray slipped to 156 in the world after only returning to action at Queen's last week following 11 months out with a hip injury.
Andy Murray won 6-1, 6-3 in the south coast English town of Eastbourne in Monday during a warm-up event for Wimbledon.
Murray was playing only his second match after 11 months on the sidelines with a hip injury that required surgery in January.