Cilic, seeded 14th, has endured an indifferent season due partly to injury. He'll be hoping to see a return to form on North American hard courts, where he won the US Open in 2014.
The Croatian earned his first victory in the Francophone city since 2011 in the tournament which alternates annually with Toronto.
Cilic never faced a break point in his 98-minute win, despite producing just two aces from his usually reliable serve.
The straight-sets success was welcome for Cilic, who improved to 14-11 in 2019.
"I didn't know what to expect," he said after the first-time meeting with big-serving Klahn. "I'm glad I played the first set really well.
"In the second he started to play better and was serving big bombs; it got a lot tougher.
"I'm glad we didn't go into a third set because then you never know... I'm happy with this win."
Cilic, Wimbledon runner-up in 2017 at a 2018 Australian Open finalist, was bothered in the first part of the season by a right knee injury.
He had to fight his way to victory in the second-set tiebreaker after losing a 3-0 lead in the decider.
Cilic also annulled a Klahn set point which would have forced a deciding set.
Felix Auger-Aliassime won the highly-touted battle of Canadians over Vasek Pospisil 6-2, 6-7 (3/7), 7-6 (7/3).
An electrifying clash lasting just over two and a half hours marked another success in a breakthrough season for the 18-year-old winner.
The world number 21 has already reached his first three career ATP singles finals and three months ago became the youngest player to crack the Top 25 since Lleyton Hewitt in 1999.
He next faces another Canadian, Milos Raonic, the 17th seed, whom he beat at Stuttgart in June.
In other first-round action, tournament debutant Dan Evans of Britain booked a clash with top-seeded Rafael Nadal with a 6-4, 7-6 (8/6) victory over Australian Alex de Minaur.
De Minaur was broken twice, unable to profit from 10 aces against his 53rd-ranked opponent who emerged from qualifying rounds.
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