After easing through the opening set, the world no.1 dropped the second after losing four games in a row. The 27-year-old recovered his bearings to see out the contest 6-4 4-6 6-2 6-0 and set up a second-round tie against Martin Klizan of Slovakia.
Murray's preparations for the second major of the year have been disrupted by an elbow injury and illness. After Angelique Kerber's shock exit in the opening round on Sunday (28 May), all eyes were on the Scot to see if he could clear the first hurdle.
The opening set was relatively straightforward for Murray, who moved ahead after 45 minutes before allowing his opponent back into the contest as Kuznetsov secured two breaks of serve.
Murray then rediscovered the verve that has been missing from his game so often in 2017, moving through the gears in the third and dropping just two games before administering a whitewash in the fourth.
Fellow Briton Kyle Edmund also progressed through to the second round of competition at Roland Garros for the third year running with an impressive win over Gastao Elias of Portugal.
Once again relishing the challenge of playing on clay, the British no.2 secured a straight sets victory over his opponent, wrapping up a 6-3 6-2 7-5 win in under two hours.
Elsewhere on day three of competition, former champion Stanislas Wawrinka eased into the second round with a straight sets win over Jozef Kovalik, ranked no. 152 in the world. Juan Martin del Potro marked his return to the competition by seeing off fellow Argentine Guido Pella, Nick Kyrgios defeated Philipp Kohlschreiber and Kei Nishikori beat Thanasi Kokkinakis.
There were also wins for women's tournament favourite Simona Halep, Elina Svitolina, Gael Monfils, John Isner and Eugenie Bouchard. Johanna Konta was upset by world no.109 Hsieh Su-Wei, while ninth seed and Italian open champion Alexander Zverev lost in four sets to Fernando Verdasco.