The second seed had been trailing by two sets to one but up a break in the fourth when play was suspended on Monday due to darkness.
The Scot duly returned to complete the job after making the crucial breakthrough at 5-all in the decider to set up a second-round encounter with 164th-ranked French wildcard Mathias Bourgue.
"He's always been extremely difficult to play," said Murray after a ninth career comeback from two sets down.
"He was playing drop shots, hitting the ball very flat, it was very difficult to get into a rhythm. That's credit to him and the way he played."
While Murray toiled, there were no such problems for nine-time champion Rafael Nadal who extended his record at Roland Garros to 71 wins against just two losses.
The fourth-seeded Spaniard made quick work of 100th-ranked Sam Groth, taming the Australian's powerful serve with eight breaks in a resounding 6-1, 6-1, 6-1 win.
However, Kerber became the tournament's first major casualty as Dutchwoman Bertens consigned the German third seed to a 6-2, 3-6, 6-3 defeat.
"The first rounds are always tough in the tournament, especially for me, but, yeah, what can I say? It happens," lamented Kerber, who was hampered by an ongoing shoulder problem.
"She's a tough opponent. I was ready. But, yeah, at the end she made the important points."
Czech seventh seed Tomas Berdych, a former French Open semi-finalist, cruised to a straight-sets win over Canada's Vasek Pospisil.
Austria's Dominic Thiem struggled initially against Inigo Cervantes, dropping the opening set to the Spaniard, but the 13th seed rallied to claim a 3-6, 6-2, 7-5, 6-1 victory.
Thiem, who retained his Nice title over the weekend, is the only player alongside Nadal with multiple clay-court tournament wins this season.
John Isner fired 40 aces to advance 6-7 (4/7), 7-6 (14/12), 7-6 (9/7), 7-5 over Australian John Millman, while Bernard Tomic put recent struggles behind him to dispatch American Brian Baker in three sets.
Swiss eighth seed Timea Bacsinszky swept past Spanish lucky loser Silvia Soler-Espinosa 6-3, 6-1, while 12th seed Carla Suarez Navarro needed three sets to beat Czech qualifier Katerina Siniakova.
Former runner-up Sam Stosur overcame Misaki Doi of Japan 6-2, 4-6, 6-3, but Francesca Schiavone, the 2010 champion, bowed out in straight sets to French 26th seed Kristina Mladenovic.
Jelena Jankovic also made an early exit, the Serbian 23rd seed dumped out by Tatjana Maria as the German won for the first time in Paris.
Later, world number one Novak Djokovic was launching another bid to complete a career Grand Slam with Taiwan's Lu Yen-Hsun the world number 95, his first-round opponent.
Women's defending champion Serena Williams, who is one Grand Slam title shy of Steffi Graf's Open era record of 22, was also due to open her campaign against 77th-ranked Slovak Magdalena Rybarikova.