Shved, who joined the club only five days ago from Celtic, was handed the first goal when Leipzig goalkeeper Peter Gulacsi miskicked the ball outside the box early on, allowing the winger to tap it into the empty net.
The Germans upped the pressure after the break and it paid off in the 57th minute when Mohamed Simakan completed a quick move for the equaliser.
But their joy barely lasted a minute when Shved, set up by Mykhailo Mudryk, this time benefited from a deflection to beat Gulacsi once more.
Mudryk then turned scorer in the 76th to put the game to bed before delivering another superb assist for Lassina Traore to add a fourth goal five minutes from the end.
With coach Domenico Tedesco under mounting pressure, Leipzig, who have won just one of their five Bundesliga games so far, travel to Real Madrid next week. Shakhtar face Celtic.