Djokovic, a three-time Wimbledon champion, was leading 6 -3, 2-0 when Slovakian world number 47 Klizan retired after just 40 minutes on Centre Court due to a calf problem.
The 30-year-old Serb will face Adam Pavlasek of the Czech Republic for a place in the last 32.
"He had issues walking on to court. I tried to focus on my game plan, I was serving well and when it mattered I made a break," said Djokovic, who is aiming to win a first Grand Slam title since last year's French Open.
It was the kind of trouble-free first round Djokovic would have been hoping for after the most turbulent period of his career.
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The world number four crashed out in the third round at Wimbledon last year and has struggled badly in 2017, the 12- time major winner losing in the Australian Open second round and the French Open quarter-finals.
With defending champion Serena Williams sidelined while she prepares to give birth to her first child, the race to win the women's title is the most wide open in a generation.
The German has failed to maintain that form this season and her poor run culminated in an embarrassing French Open first round loss to Ekaterina Makarova last month.
But the top seed showed glimpses of her best form as she saw off American qualifier Irina Falconi 6-4, 6-4 in 87 minutes on Centre Court.
"I'm happy to be back, but this year is completely different for me. I'm just happy I'm through the first round," Kerber said.
Big-serving Raonic became the first Canadian man to make a Grand Slam final with his impressive Wimbledon run 12 months ago before losing the title match to Andy Murray.
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The sixth seed made a business-like start to his bid to make a second major final, in the process extended German world number 53 Struff's losing streak at Grand Slams to 10 matches.
French 22nd seed Richard Gasquet slumped to a 6-3, 6-4, 5-7, 6-2 loss against Spain's David Ferrer.
Argentina's Juan Martin Del Potro, a semi-finalist in 2013, defeated Australian Thanasi Kokkinakis 6-3, 3-6, 7-6 (7/2), 6-4 as the 29th seed moved towards a potential third round classic against Djokovic.
Australia's Bernard Tomic said he had lost his "respect" for tennis following his lacklustre 6-4, 6-3, 6-4 loss to Mischa Zverev.
"I wasn't there with my mental state to perform," Tomic said.
"I don't know why but I felt a little bit bored out there, to be completely honest."
Gael Monfils, the flamboyant French 15th seed, crushed German qualifier Daniel Brands 6-3, 7-5, 6-4.
Australian qualifier Arina Rodionova, ranked 166th, saved seven match points before defeating Russian 16th seed Anastasia Pavlyuchenkova 3-6, 7-6 (8/6), 9-7.
Later on Tuesday, Roger Federer, chasing a record eighth Wimbledon title, starts his 19th All England Club campaign against Ukrainian world number 84 Alexandr Dolgopolov.
Having unexpectedly won the Australian Open in January before taking the clay-court season off to rest, Federer can lay down a marker to his title rivals by claiming a record 85th match win at Wimbledon.