The Serb is playing a pre-Wimbledon event for the first time since 2010 and will next face the winner from a pair of Americans, either Donald Young or Jared Donaldson.
The three-time Wimbledon champion got onto court after rain washed out Tuesday's play -- save one game in the Djokovic match -- leaving organisers with a backlog to get through.
The former world number one claimed his fifth victory in as many attempts against Pospisil, who is now 4-22 lifetime against top 10 opponents.
"Yesterday it obviously wasn't really enjoyable to see the rain falling all day.
"But today we almost played a full match, so I'm glad for that. I felt good on the court, considering it's the first match.
"I played against an opponent that has a really good game for grass. Serves well, comes to the net, has a good variety.
"It was a really, really solid win. I'm happy with it."
In the men's first round, Russian Daniil Medvedev knocked out fifth seed Sam Querrey, beating the American 6-3, 7-5. German Mischa Zverev, seeed sixth, beat Ryan Harrison of the US 6-4, 7-6 (11/9).
On the women's side, top-ranked Angelique Kerber recovered to beat Czech Kristyna Pliskova 4-6, 6-1, 7-5, as the German returned to action for the first time since her opening round defeat at the French Open.
Kerber is fighting to hold onto her number one ranking and withdrew from last week's Birmingham event with a hamstring strain.
Roland Garros finalist Simona Halep, who trails Kerber by just 115 ranking points, rallied to defeat China's Duan Ying- Ying 3-6, 6-3, 6-2, while Bulgarian Tsvetana Pironkova -- a 2010 Wimbledon semi-finalist -- thumped Monica Niculescu of Romania 6-0, 6-4.
Two-time Grand Slam winner and seventh seed Svetlana Kuznetsova overcame German Mona Barthel 4-6, 7-5, 6-2 while France's Kristina Mladenovic, seeded ninth, dusted aside China's Zhang Shuai 6-0, 6-4.
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