Cilic, seeded 14th, has suffered through an indifferent season due partly to injury, but could be back in his element on hardcourt after winning the US Open in 2014.
He earned his first victory in the Francophone city since 2011 at the tournament which alternates annually with Toronto.
The Croatian never faced a break point in his 98-minute win but was able to produce only 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, runner-up in 2017 at Wimbledon and finalist at the 2018 Australian Open, was bothered in the first part of the season by a right knee injury.
The seed 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.
Australian Alex de Minaur fell 6-4, 7-6 (8/6) to tournament debutant Dan Evans, sending the British winner into a second-round challenge against defending champion and ATP number two Rafael Nadal.
De Minaur lost serve twice and was unable to profit from 10 aces against his 53rd-ranked opponent who emerged from qualifying rounds.