The Brazilian playmaker joined Shakhtar from Benfica last month and scored the winner with a sublime first-half finish in the principality.
The 23-year-old Pedrinho, who signed for the Ukrainians for 18 million euros ($21 million), repaid the faith showed in him by Shakhtar coach Roberto De Zerbi in the 19th minute.
He picked up the ball on the edge of the box, dribbled past three defenders before coolly finishing past goalkeeper Radoslaw Majecki with his weaker right foot.
The hosts had two good chances to draw level before the break but captain Wissam Ben Yedder and midfielder Aurelien Tchouameni failed to beat Anatolii Trubin.
Trubin, 20, who was an unsued member of Ukraine's Euro 2020 squad, was kept busy in the second half but kept out efforts by winger Aleksandr Golovin, Ben Yedder and substitute Myron Boadu.
Elsewhere, American attacking midfielder Brenden Aaronson scored in the 90th minute as Salzburg came from behind to beat Brondby 2-1.
Mali forward Adama Traore claimed a double as Sheriff Tiraspol, who could become the first Moldovan club to reach the group stage, hammered Dinamo Zagreb 3-0.
On Wednesday, two-time champions Benfica host 1988 winners PSV Eindhoven, Malmo welcome Bulgarians Ludogorets Razgrad to Sweden and Swiss Super League holders Young Boys face their Hungarian counterparts Ferencvaros.