Home favourite Kuznetsova – who won the WTA Tour tournament in 2015 before returning to successfully defend her title the after year – managed just 13 winners as Muchova triumphed 6-2 6-3 to set up a second-round clash with fourth seed Donna Vekic.
Anastasija Sevastova had no such problems in her opening encounter, easing past Aliaksandra Sasnovich 6-1 7-5, while Anna Kalinskaya overcame Anastasia Potapova 6-0 3-6 6-4 via a tense final set that saw the two players combine to produce seven breaks of serve.
Elina Svitolina and Kiki Bertens are the top seeds for the tournament, though neither are in action during Tuesday's play in Russia.
At the Luxembourg Open, Viktoria Kuzmova rallied from a set down to defeat teenager Katie Volynets 6-7 (4-7) 7-6 (7-1) 6-1.
The sixth seed let slip a 5-2 lead in the second set before eventually prevailing in a tie-break, while she dropped just 10 points in the decider to triumph after two hours, 33 minutes on court.
Margarita Gasparyan, Tatjana Maria and Denisa Allertova also recorded three-set wins in the main draw, the latter coming through via a tense tie-break against Mandy Minella.