Wimbledon champion Djokovic, 31, was due to face South Korea's Hyeon Chung, who beat him at the Australian Open earlier this year, in the first round but the rising star had to withdraw due to a back injury.
Basic came in to replace Chung but he had no answer to ninth seed Djokovic in the first set in Toronto as the Serb broke him in game six and then served out.
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The Bosnian fought back in the second set as the pair exchanged two breaks of serve each which led to a tie break.
In the tie break Djokovic held his nerve to secure progression to the next round, where he will face Peter Polansky.
"I've never played Basic. I've known him, I've practiced with him, but I've never played an official match so definitely not easy to find a rhythm from the beginning of the match," said Djokovic. "There were frustrations at times but that means I care."
The Serbian world number 10 was not as sharp as he will need to be if he hopes to triumph here for the first time since 2016, but it was enough on a day when Basic was misfiring on both his forehand and serve.
In other first-round action, Swiss wild card Stan Wawrinka upset 16th-seeded Australian Nick Kyrgios 1-6 7-5 7-5 while Dutchman Robin Haase beat Japan's Kei Nishikori 7-5 6-1.
ROGERS CUP FIRST ROUND RESULT
Benoit Paire beat Jared Donaldson 6-3, 6-4
Marton Fucsovics beat Jaoa Sousa 6-4, 7-6(2)
Stan Wawrinka beat Nick Kyrgios 1-6, 7-5, 7-5
Diego Schwartzman beat Kyle Edmund 6-1, 6-2
Sam Querrey beat Adrian Mannarino 6-2, 7-5
Borna Coric beat Vasek Pospisil 6-4, 6-3
Robin Haase beat Kei Nishikori 7-5, 6-1
Fabio Fognini beat Steve Johnson 6-4, 6-4
Pablo Carreno Busta beat Yoshihito Nishioka 6-0 RET
Karen Khachanov beat Filip Krajinovic 6-3, 6-2
Pierre-Hugues Herbet beat Albert Ramos-Vinolas 4-6, 6-3, 6-4
Fernando Verdasco beat Peter Gojowczyk 6-2, 6-2
Frances Tiafoe beat Marco Cecchinato 7-6(3), 6-1
Milos Raonic beat David Goffin 6-3, 6-4
Ryan Harrison beat Mackenzie McDonald 6-4, 6-2
Ilya Ivashka beat Yuichi Sugita 6-2, 6-3
Stefanos Tsitsipas beat Damir Dzumhur 6-3, 7-6(3)
Peter Polansky beat Matthew Ebden 7-6(3), 6-4
Novak Djokovic beat MIrza Basic 6-3, 7-6(3)
Daniil Medvedev beat Jack Sock 6-3, 3-6, 6-3
Felix Auger-Aliassime beat Lucas Pouille 6-4, 6-3
Bradley Klahn beat David Ferrer 7-6(5), 6-4
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